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http://www.archive.org/details/descriptivegeomeOOmillrich 


DESCRIPTIVE  GEOMETRY 


BY 

ADAM  V.  MILLAR 

ASSISTANT  PROFESSOR  OF  DRAWINO  UNIVERSITY  OF  WISCONSIN 
AND 

EDWARD  S.  MACLIN 

INSTRUCTOR  IN  DRAWING  UNIVERSITY  OF  WISCONSIN 


McGRAW-HILL  BOOK    COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1913 


Copyright,  1913 

BY 

A.  V.  MILLAR 

AND 

E.  S.  MACLIN 


STATE  JOURNAL  PRINTING  COMPANY 

Printers  and  Sterkotypers 
madison,  wis. 


PREFACE 


In  preparing  the  following  text  in  descriptive  geometry,  the 
authors  have  endeavored  (1)  to  make  the  subject  easier  for 
the  student,  (2)  to  help  the  student  to  visualize  magnitudes  in 
space,  and  (3)  to  present  the  subject  more  nearly  in  accord 
with  commercial  practice. 

In  order  to  accomplish  these  three  things  the  ground  line  is 
omitted.  When  the  projections  of  several  points  are  given 
without  the  ground  line  being  shown,  the  distances  of  the 
points  from  the  horizontal  or  vertical  planes  of  projection  are 
not  determined.  The  vertical  projections,  however,  do  show 
the  relative  heights  of  the  points  and  the  horizontal  projections 
show  the  relative  distances  of  the  points  from  the  vertical  plane. 
It  is  the  relative  distances  of  points  of  an  object  from  a  plane 
with  which  we  are  concerned,  since  the  distance  of  the  whole 
object  from  the  plane  of  projection  does  not  change  the  or- 
thographic projection  of  the  object  on  that  plane. 

When  it  is  desired  to  locate  points  which  are  given  distances 
from  the  planes  of  projection,  the  ground  lioe  must  be  used. 
Even  when  the  ground  line  is  not  shown,  it  is  understood  to  be 
at  right  angles  to  the  line  joining  the  two  projections  of  the 
same  point. 

By  the  omission  of  the  ground  line,  and  therefore  the  traces 
of  a  plane,  the  student's  attention  is  centered  on  the  object  or 
magnitude  in  space  and  not  on  the  planes  of  projection.  This 
teaches  the  student  to  visualize  the  object  rather  than  mem- 
orize the  projections  of  the  object.  The  subject  is  thus  made 
easier  because  memorizing  constructions  and  keeping  the  draw- 
ing rather  than  the  object  in  mind  are  the  greatest  hinderances 
which  the  student  encounters  in  mastering  the  subject.  Since 
the  ground  line  is  omitted  in  commercial  work,  the  subject 


259669 


IV  PREFACE 

taught  in  the  above  maimer  is  more  in  accord  with  that  prac- 
tice. 

The  third  quadrant  is  used  quite  generally  in  the  drafting 
offices  of  this  country.  It  seems  logical,  therefore,  to  present 
the  subject  of  descriptive  geometry  in  the  third  quadrant,  which 
is  done  in  the  present  text.  Since  in  this  text  there  is  no 
particular  horizontal  or  vertical  plane,  objects,  such  as  cones 
and  cylinders,  can  be  placed  in  their  natural  positions  rather 
than  being  inverted  to  bring  their  bases  in  a  horizontal  plane  of 
projection.  This  removes  one  of  the  chief  objections  to  the 
third  quadrant. 

Although  the  method  used  here  in  presenting  the  principles 
of  descriptive  geometry  is  new  in  American  texts  on  the  sub- 
ject, it  is  used  to  some  extent  by  French  authors  such  as  Javary, 
'*F.  J.,''  and  others.  It  has  been  used  at  the  University  of  Wis- 
consin for  some  time  and  the  results  have  been  most  satisfac- 
tory. 

The  authors  have  consulted  many  descriptive  geometries  in 
preparing  the  following  text  but  are  particularly  indebted  for 
suggestions  and  problems,  to  the  following :  Phillips  and  Millar, 
Fishleigh,  Ames,  Bartlett,  and  MacCord. 


CONTENTS 

Page 

Introduction  1 

CHAPTER  I 

FIRST  PRINCIPLES 

Projection  5 

Orthographic  Projection 6 

Picturing  Magnitudes  in  Space 8 

Points 9 

Lines 12 

Planes 16 

Revolution  and  Counter-Revolution 19 

Line  Conventions 24 

CHAPTER  II 

THE  ELEMENTARY  PRINCIPLES   OF  THE  POINT,   STRAIGHT 
LINE,  AND  PLANE 

To  Find  the  Length  of  a  Line  and  the  Angles  Which  it  Makes 

V7ITH  H  AND  V 26 

To  Find  the  Projections  of  a  Line  Which  Makes  a  Given  Angle 

vriTH  H  OR  V 27 

To  Find  the  Angle  Which  a  Given  Plane  Makes  with  H  or  V 28 

To  Represent  a  Plane  Which  Makes  a  Given  Angle  with  H  or  V  30 

To  Find  the  Angle  Between  Two  Lines 31 

To  Find  a  Line  Which  Contains  a  Given  Point  and  Makes  a 

Given  Angle  with  a  Given  Line 33 

To  Represent  a  Plane  Which  Contains  a  Given  Point  and  is 

Parallel  to  Two  Given  Lines 35 

To  Find  the  Point  in  Which  a  Line  Pierces  a  Plane 36 

To  Find  the  Line  of  Intersection  of  Two  Planes 37 

A  Line  Perpendicular  to  a  Plane 40 

To  Find  the  Distance  From  a  Point  to  a  Plane 40 

To  Find  a  Point  Which  is  a  Given  Distance  From  a  Plane 41 

To  Represent  a  Plane  Which  Contains  a  Point  and  is  Perpen- 
dicular TO  A  Line 42 


VI  CONTENTS 

Page 

To  Find  the  Projection  of  a  Line  on  a  Plane 43 

To  Find  the  Angle  Which  a  Line  Makes  with  a  Plane 44 

To  Find  the  Angle  Between  Two  Planes 45 

To  Find  the  Common  Perpendicular  to  Two  Lines 46 

Auxiliary  Planes  of  Projection 48 

Problems  Intolvinq  Points,  Lines,  and  Planes 50 

CHAPTER  III 

APPLICATIONS  OF  THE  ELEMENTARY  PRINCIPLES  OF  THE 
POINT,  STRAIGHT  LINE.  AND  PLANE 

Shades  and  Shadows 57 

Plane  Sections  and  Developments  of  the  Surfaces  of  Prisms 

AND  Pyramids 64 

Intersections  of  the  Surfaces  of  Prisms  and  Pyramids 70 

CHAPTER  IV 

CURVED  LINES  AND  SURFACES 

Generation  and  Classification  of  Lines 77 

Projections  of  Curves 78 

Tangents  and  Normals  to  Lines 79 

Curves  of  Single  Curvature 80 

Curves  of  Double  Curvature 85 

Generation  and  Classification  of  Surfaces 86 

Surfaces  of  Revolution 87 

Tangent  Planes  to  Surfaces.    Normal  Lines  and  Planes 89 

Single  Curved  Surfaces 92 

Warped  Surfaces 99 

Double  Curved  Surfaces 109 

Problems  on  Tangent  Planes  to  Surfaces 113 

CHAPTER  V 

PLANE  SECTIONS  AND  DEVELOPMENTS  OF  CURVED  SURFACES 

Right  Cylinder 115 

Oblique  Cylinder   117 

Oblique  Cone  118 

Plane  Section  of  a  Warped  Surface 120 

Plane  Section  of  a  Surface  of  Revolution 120 


CONTENTS  Vll 

Page 
Plane  Section  of  Any  Surface  by  the  Use  of  an  Auxiliary  Plane 

OF  Projection  122 

Problems  on  the  Plane  Sections  of  Surfaces 124 


CHAPTER  VI 

INTERSECTIONS  OF  CURVED  SURFACES 

General  Method  for  Finding  the  Line  of  Intersection  of  Two 

Surfaces    125 

Two  Oblique  Cylinders 125 

Cone  and  Cylinder 127 

To  Determine  in  Advance  the  Nature  of  the  Line  of  Intersec- 
tion      127 

Two  Cones 129 

Sphere  and  Cone  or  Sphere  and  Cylinder 129 

Problems  on  the  Line  of  Intersection  of  Surfaces 130 


INTRODUCTION 


It  may  help  some  instructors  who  contemplate  using  the  fol- 
lowing text-book  to  know  how  the  authors  have  used  the  book 
in  their  classes.  With  this  in  view,  the  following  general 
method  for  conducting  the  course  is  suggested  and  an  outline 
of  lessons  given.  Each  instructor  will,  no  doubt,  need  to  alter 
the  outline  to  some  extent  to  suit  the  conditions  under  which 
he  works. 

At  the  University  of  Wisconsin,  descriptive  geometry  is 
given  as  a  three  credit  course  for  one  semester  of  eighteen 
weeks.  Each  week's  work  consists  of  one  general  lecture  for 
all  students  in  the  course,  one  recitation,  and  two  two-hour 
drafting  periods  for  each  section.  One  of  the  two-hour  draft- 
ing periods  is  sometimes  turned  into  a  one-hour  recitation 
period. 

At  the  lecture,  the  general  principles  involved  in  the  next 
lesson  are  explained,  general  announcements  made,  and  prob- 
lems assigned  for  a  home  plate  which  is  to  be  handed  in  at  tho 
beginning  of  the  recitation  period.  At  the  recitation,  the  stu- 
dents are  drilled  in  the  analyses  of  the  problems  and  then  sent 
to  the  black  board  with  some  particular  problem  to  solve. 

In  the  drafting  room,  each  student  is  given  a  slip  similar  to 
the  following: 

Plate  1 

Article  10,  Problem    4 

''  10,       ''         14 

''  15,       ''  5 

''  15,       ''         17 

As  far  as  possible  duplicate  slips  are  avoided.  The  student 
solves  the  problems  and  if  he  has  time  letters  the  statements  of 


INTRODUCTION 


the  problems.  The  work  is  done  on  a  ll"xl5"  sheet  and  is  left 
in  pencil.  Neatness,  clearness,  and  accuracy  are  demanded. 
With  the  exception  of  a  few  plates,  the  work  is  completed  and 
handed  in  at  the  close  of  each  two-hour  drafting  period.  The 
plates  are  then  corrected,  graded,  and  returned  to  the  student 
at  the  next  drafting  period.  This  method  of  giving  a  plate  to 
be  completed  each  time  the  student  comes  to  the  drafting  room 
has  the  following  advantages:  the  student  comes  better  pre- 
pared for  his  work,  he  wastes  no  time  in  the  drafting  room,  and 
the  attendance  is  improved. 

Unannounced  written  quizzes  are  given  in  the  drafting  room 
about  every  three  weeks.  Neatness  and  clearness  of  the  con- 
struction counts  for  15%  of  the  grade. 

The  following  outline  gives  the  lesson  assignments  for  the 
recitation  work. 


Lesson  1. 

Articles  1 

to 

12  incl 

u 

''       2. 

13 

19 

''      .3. 

20 

26 

4. 

27 

30 

''       5. 

31 

34 

''       6. 

35 

41 

''   7. 

42 

45 

"   8. 

46 

51 

''       9. 

52 

55 

''     10. 

56 

58 

''  11. 

59 

61 

**  12. 

62 

68 

''     13. 

69 

73 

''     14. 

74 

76 

''     15. 

77 

79 

''     16. 

80 

95 

*'  17. 

96 

107 

''     18. 

108 

118 

''     19. 

119 

128 

''     20. 

129 

136 

/ 

INTRODUCTION 

SOI 

1  21. 

Articles  137  to  152  inclusive 

a 

22. 

153''  158      '' 

(  c 

23. 

159''  161      " 

In  case  the  time  allowed  for  descriptive  geometry  is  not 
enough  to  permit  the  giving  of  the  course  as  outlined  above,  it 
is  suggested  that  shades  and  shadows  and  the  latter  part  of 
warped  surfaces  be  omitted. 


DESCRIPTIVE  GEOMETRY 


CHAPTER  I 
FIRST  PRINCIPLES 

1.  Projection.  If  from  a  point  S,  Fig.  1,  straight  lines  are 
drawn  through  a  series  of  points  A,  B,  C  . . . ,  the  points  a,  h,  c 
...  in  which  these  lines  pierce  the  plane  T,  are  the  projections 
of  the  points  A,  B,  C on  this  plane. 

S  is  called  the  point  of  sight. 

A,  B,  C,  ...  are  points  of  a  magnitude  or  object  in  space. 

Sa,  Sh,  So  . . .  are  projecting  lines  of  the  points  A,  B,  C, 

T  is  the  plane  of  projection. 


/b/  / 

/  ^  / 


A\ 


/n^/j^j\/i 


Fig.  1. — Perspective. 


Fig.  2. — Oblique 
projection. 


%i 


c 


7M\ 


F^G.  3. — Orthographic 
projection. 


Leaving  color  out  of  consideration,  the  projections  a,  6,  c, 
. . .  present  the  same  appearance  to  the  eye,  situated  at  the 
point  of  sight,  as  the  points  A,  B,  C,  ...  in  space. 

In  perspective,  Fig.  1,  the  point  of  sight  is  at  a  finite  distance 
from  the  plane  of  projection.     The  projecting  lines  diverge. 

In  oblique  and  orthographic  projection.  Figs.  2  and  3,  the 
point  of  sight  is  at  an  infinite  distance  from  the  plane  of  pro- 
jection.    The  projecting  lines  are  parallel. 


DESCRIPTIVE   GEOMETRY 


The  projection  is  oblique  when  the  projecting  lines  are  par- 
allel to  each  other  and  oblique  to  the  plane  of  projection. 

The  projection  is  orthographic  when  the  projecting  lines  are 
perpendicular  to  the  plane  of  projection. 

ORTHOGRAPHIC  PROJECTION 

2.  A  point  in  space  is  not  completely  determined  by  its  or- 
thographic projection  on  one  plane  for  the  distance  of  the  point 
from  the  plane  is  not  shown  by  its  projection.  All  points  A,  B, 
C,  Fig.  4,  which  lie  in  a  vertical  straight  line  have  the  same 
projection  on  a  horizontal  plane.  There  are  two  methods  of 
representing  definitely  a  point  in  space.  One  method  is  to  give 
J  its  projection  on  a  plane  and  also 

A*  its  distance  from  that  plane.     This 

ig  method  is  used  in  making  contour 

maps;  a  contour  line  being  a  line 
joining  the  projections  of  all  points 
which  are  a  given  distance  above  or 


Ct 


; V  wnicn  are  a  given  aisiance  aoove  or 

^^^  \l  below  a  base  plane.  The  other  method 
"  is  to  use  two  (or  more)  different 
Fig.  4. — a,  6,  c,  projections  of  planes  of  projection.  When  two  pro- 
points  A,  B,  C,  on  plane  T.  jections  are  used,  one  is  usually  con- 
sidered the  principal  projection.  The  other  is  supplementary 
showing  the  distance  of  the  point  from  the  plane  upon  which  the 
principal  projection  is  made.  The  method  which  uses  the  two 
projections  of  a  point  is  the  more  common. 

3.  PlaJies  of  projection.  In  orthographic  projection,  two 
planes  are  generally  used,  Fig.  5,  one  horizontal  and  the  other 
vertical,  called  respectively: 

(a)  The  horizontal  plane  of  projection,  or  H. 

(b)  The  vertical  plane  of  projection,  or  V. 

Their  intersection  is  called  the  ground  line,  or  G.  L. 

Sometimes  other  planes  of  projection  are  used  which  are  per- 
pendicular to  either  H  or  V.  If  the  plane  of  projection  is  per- 
pendicular to  both  H  and  V  and  therefore  perpendicular  to  the 
ground  line,  it  is  called  a  profile  or  end  plane,  or  P. 


ORTHOGRAPHIC  PROJECTION 


Point  of  sight.  The  point  of  sight  for  the  horizontal  plane  is 
an  infinite  distance  above  H  and  for  the  vertical  plane  it  is  an  in- 
finite distance  in  front  of  V.  In  the  horizontal  view,  the  ground 
line  represents  the  vertical  plane  seen  edgewise,  and  in  the 
vertical  view,  the  ground  line  represents  the  horizontal  plane 
seen  edgewise. 

4.  The  four  quadrants.  The  right  dihedral  angles  formed  by 
the  intersection  of  the  horizontal  and  vertical  planes  of  projec- 
tion are  known  as  the  first,  second,  third,  and  fourth  quadrants, 
Fig.  5. 

The    first    quadrant    is 
above  H  and  in  front  of  V. 
The  second  quadrant  is 
above  H  and  back  of  V. 

The  third  quadrant  is 
below  H  and  back  of  V. 

The  fourth  quadrant  is 
below  H  and  in  front  of  V. 
5.  The  drawing.    In  or- 
der to  represent  both  the 
horizontal      and     vertical 

projections  of  an  object  on 
Fig.  5.-Principal  planes  of  projection.       ^^^  ^^^^  ^^^^^   ^^  p^p^^.^ 

the  planes  of  projection  must  be  brought  together.  This  is 
accomplished  by  keeping  one  of  the  planes  fixed  and  revolving 
the  other  about  the  ground  line  as  an  axis  until  the  two  planes 
coincide.  If  the  horizontal  plane  is  kept  fixed,  the  upper  part 
of  the  vertical  plane  is  revolved  backward  until  it  coincides 
with  the  back  part  of  the  horizontal  plane.  This  wiU  be  found 
convenient  if  the  work  is  done  on  a  drafting  board  which  is  in  a 
horizontal  position.  If  the  vertical  plane  is  kept  fixed,  the 
front  part  of  the  horizontal  plane  is  revolved  downward  until  it 
coincides  with  the  lower  part  of  the  vertical  plane.  This  will 
be  found  convenient  when  working  on  the  blackboard.  By 
either  method,  the  first  and  third  quadrants  are  opened  and  the 
second  and  fourth  closed. 


8  DESCRIPTIVE   GEOMETRY 

When  the  profile  or  end  plane  is  used,  it  is  brought  into  the 
vertical  plane  by  revolving  it  about  its  intersection  vrith  the 
vertical  plane. 

It  must  be  remembered  that  the  projections  are  made  vrhile 
the  planes  of  projection  are  still  at  right  angles,  the  planes  be- 
ing brought  together  simply  for  convenience  in  making  all  of 
the  projections  on  one  sheet  of  paper. 

PICTURING  MAGNITUDES  IN  SPACE 

6.  In  order  to  solve  problems  intelligently,  the  student  should 
learn  to  picture  to  himself  points,  lines,  and  objects  in  their 
proper  positions  v^ith  reference  to  the  planes  of  projection. 
When  the  dravring  is  in  a  horizontal  position,  it  is  best  to  con- 
sider the  plane  of  the  dravring  as  the  horizontal  plane  of  pro- 
jection. The  vertical  plane  of  projection  should  be  pictured  as 
a  plane  vrhich  contains  the  ground  line  and  is  perpendicular  to 
the  plane  of  the  drawing.  The  student  should  place  one  of  his 
triangles  to  represent  the  vertical  plane.  The  other  triangle 
when  placed  at  right  angles  to  the  horizontal  and  vertical  planes 
of  projection  will  represent  the  profile  or  end  plane. 

The  first  quadrant  is  above  the  drawing  and  in  front  of  the 
ground  line.  The  second  quadrant  is  above  the  drawing  and 
behind  the  ground  line.  The  third  quadrant  is  below  the  draw- 
ing and  behind  the  ground  line.  The  fourth  quadrant  is  below 
the  drawing  and  in  front  of  the  ground  line. 

The  magnitude,  such  as  a  point,  line,  or  object,  is  then  pic- 
tured in  its  proper  position  with  reference  to  the  planes  of  pro- 
jection. The  top  or  horizontal  view  is  obtained  by  looking 
straight  down  from  above,  the  front  or  vertical  view  by  look- 
ing from  the  front,  and  the  end  or  profile  view  by  looking 
through  the  end  plane.  After  these  projections  are  made,  the 
planes  of  projection  are  brought  together  by  the  method  ex- 
plained in  Art.  5.  The  magnitudes  should  always  be  pictured 
in  space  before  the  drawing  is  made. 


POINTS 


POINTS 


7.  Let  A,  Fig.  6,  be  any  point  in  space  and  H,  V,  and  P  the 
planes  of  projection.  The  line  Aa,  through  the  point  and  per- 
pendicular to  the  horizontal  plane,  is  the  horizontal  projecting 
line  of  the  point  and  its  intersection  a,  with  the  horizontal  plane, 
is  the  horizontal  projection  of  the  point.  Similarly  the  line  Aa\ 
through  the  point  and  perpendicular  to  the  vertical  plane,  is  the 
vertical  projecting  line  of  the  point  and  its  intersection  a'  with 
the  vertical  plane  is  the  vertical  projection  of  the  point.  In 
like  manner  the  line  A.a",  through  the  point  and  perpendicular 
to  the  profile  plane,  is  the  profile  projecting  line  of  the  point 
and  its  intersection  a"  with  the  profile  plane  is  the  profile  pro- 
jection of  the  point. 


Fig.  6. — Point  A  in  3rd  quadrant. 


Fig.  7. 


Any  two  projections  of  a  point  determine  its  position  in 
space,  for  the  projecting  lines  intersect  in  the  only  point  which 
can  have  the  given  projections. 

Since  the  vertical  projecting  line  of  a  point  in  space  is  par- 
allel to  the  horizontal  plane,  the  distance  from  the  point  in 
space  to  the  horizontal  plane  is  equal  to  the  distance  from  its 
vertical  projection  to  the  ground  line.  Moreover,  the  horizon- 
tal projecting  line  of  a  point  in  space  is  parallel  to  the  vertical 


10 


DESCRIPTIVE   GEOMETRY 


plane,  therefore  the  distance  from  the  point  in  space  to  the  ver- 
tical plane  is  equal  to  the  distance  from  its  horizontal  projec- 
tion to  the  ground  line. 

Since  a'h  remains  perpendicular  to  the  ground  line  during  the 
revolution  of  the  vertical  plane  from  the  vertical  to  the  horizon- 
tal position,  then  when  the  vertical  and  horizontal  planes  coin- 
cide as  they  do  on  the  drawing,  Fig.  7,  a'a  is  perpendicular  to 
the  ground  line,  that  is,  the  two  projections  of  a  point  in  space 
are  in  the  same  line  perpendicular  to  the  ground  line. 

8.  Figs.  8,  9,  and  10  show  the  projections  of  points  in  the 
first,  second,  and  fourth  quadrants  respectively.     The  figures 


^a' 


Fig.  ^.— Point  A 
in  1st  Q. 


I 

I 

at 

I 

I 
I 
__L 


'Fig.  9.— Point  A 
in  2nd  Q. 


I 

I 
ai 


Fig.  l(i.— Point  A 
in  4th  Q. 


represent  the  projections  of  the  point  after  the  planes  of  pro- 
jection have  been  brought  into  coincidence.  These  points  in 
space  should  be  definitely  pictured  from  their  projections. 
"Wlien  the  horizontal  projection  is  considered  the  principal  pro- 
jection, pictufe  directly  above  or  below  it,  the  vertical  projec- 
tion giving  the  distance  from  the  horizontal  plane.  When  the 
vertical  projection  is  considered  the  principal  projection,  pic- 
ture directly  in  front  of  or  behind  it,  the  horizontal  projection 
giving  the  distance  from  the  vertical  plane. 

It  does  not  matter  in  which  quadrant  the  object  is  placed,  the 
point  of  sight  for  the  horizontal  view  is  always  above  H,  and 
for  the  vertical  view  it  is  always  in  front  of  V. 


POINTS  11 

In  the  third  quadrant,  the  point  of  sight  is  at  an  infinite  dis- 
tance to  the  right  for  the  right  end  view  and  at  an  infinite  dis- 
tance to  the  left  for  the  left  end  view. 

9.  Natation.  Points  in  space  are  designated  by  capital  let- 
ters as  A,  B,  C,  etc.  Horizontal  projections  of  points,  small 
letters  as  a,  h,  c,  etc.  Vertical  projections  of  points,  small  let- 
ters with  prime  marks  as  a\  &',  c',  etc.  Profile  projections  of 
points,  small  letters  with  double  prime  marks  as  a",  b",  d\  etc. 

Ground  line,  full. 

The  line  joining  the  projections  of  a  point  will  be  drawn  fine, 
full  in  pencil,  and  fine,  full,  red  in  ink. 

10.  Problems.  Picture  to  yourself  points  in  the  following 
positions  and  then  draw  their  projections. 

1.  In  H  V  in  front  of  V. 

2.  In  V  \\"  below  H. 

3.  V  in  front  of  V  and  2"  above  H. 

4.  W  below  H  and  \\"  behind  V. 

5.  \"  above  H  and  W"  behind  V. 

6.  \\"  below  H  and  ir  in  front  of  V. 

7.  V  above  H  and  \\"  in  front  of  V. 

8.  1"  behind  V  and  \\"  below  H. 

9.  V  in  front  of  V  and  1"  below  H. 

10.  \\"  behind  V  and  T  above  H. 

11.  In  V  and  2"  above  H. 

12.  In  H  and  W  behind  V. 

13.  In  the  third  quadrant  V  from  H  and  \\"  from  V. 

14.  In  the  first  quadrant  1^''  from  V  and  \"  from  H. 

15.  In  the  fourth  quadrant  \\"  from  H  and  \"  from  V. 

16.  In  the  second  quadrant  \"  from  V  and  \\"  from  H. 

17.  A  point  has  its  projections  coinciding  above  the  ground  line. 
Describe  the  location  of  the  point. 

18.  The  same  as  problem  17  except  that  the  projections  of  the  point 
coincide  below  the  ground  line. 


12 


DESCRIPTIVE   GEOMETRY 


LINES 

11.  Let  AC,  Fig.  11,  be  any  straight  line  in  space  and  H,  V, 
and  P  the  planes  of  projection.  The  plane  ACc,  containing  the 
line  and  perpendicular  to  the  horizontal  plane,  is  the  horizontal 
projecting  plajie  of  the  line.  Its  intersection  ac  with  the  hor- 
izontal plane  is  the  horizontal  projection  of  the  line.  Similarly 
the  plane  ACc',  containing  the  line  and  perpendicular  to  the 
vertical  plane,  is  the  vertical  projecting  plane  of  the  line.  Its 
intersection  a'c'  with  the  vertical  plane  is  the  vertical  projection 
of  the  line.  In  like  manner  the  plane  ACc",  containing  the  line 
and  perpendicular  to  the  profile  plane,  is  the  profile  projecting 
plane  of  the  line.  Its  intersection  a"c"  with  the  profile  plane  is 
the  profile  or  end  projection  of  the  line. 


Fig.  11. — Projections  of  a  line. 

Any  two  projections  of  a  straight  line  determine  its  position 
in  space,  for  the  projecting  planes  intersect  in  the  only  line 
which  can  have  the  given  projections. 

"When  the  two  projections  of  a  line  of  indefinite  length  are 
in  the  same  perpendicular  to  the  ground  line,  the  projecting 
planes  will  coincide  and  the  line  is  undetermined.  If,  however, 
the  projections  of  any  two  points  of  a  straight  line  are  given, 
the  line  is  always  determined. 


*     # 


LINES  ^  13 

12.  Projections  of  straight  lines  in  various  positions  with  ref- 
erence to  the  the  planes  of  projection. 

If  a  line  is  oblique  to  H,  V,  and  P,  the  projections  of  the  Una 
will  be  inclined  to  the  ground  line. 

If  a  line  is  parallel  to  the  ground  line,  its  H  and  V  projections 
will  be  parallel  to  the  ground  line. 

If  a  line  is  perpendicular  to  the  horizontal  plane,  its  horizon- 
tal projection  is  a  point  and  its  vertical  projection  is  a  straight 
line  perpendicular  to  the  ground  line. 

If  a  line  is  perpendicular  to  the  vertical  plane,  its  vertical 
projection  is  a  point  and  its  horizontal  projection  is  a  straight 
line  perpendicular  to  the  ground  line. 

If  a  line  is  parallel  to  the  horizontal  plane  and  oblique  to  the 
vertical  plane,  its  vertical  projection  will  be  parallel  to  the 
ground  line  and  its  horizontal  projection  will  be  inclined  to  the 
ground  line. 

If  a  line  is  parallel  to  the  vertical  plane  and  oblique  to  the 
horizontal  plane,  its  horizontal  projection  will  be  parallel  to  the 
ground  line  and  its  vertical  projection  will  be  inclined  to  the 
ground  line. 

13.  Point  on  line.  If  a  point  is  on  a  lihe  in  space,  the  hor- 
izontal projection  of  the  point  will  be  on  the  horizontal  pro- 
jection of  the  line,  the  vertical  projection  of  the  point  will  be 
on  the  vertical  projection  of  the  line,  and  the  profile  projection 
of  the  point  will  be  on  the  profile  projection  of  the  line,  Fig.  12. 
In  order  to  represent  a  point  of  a  given  line  which  lies  in  a 
profile  plane,  use  the  profile  projection  of  the  line. 

Intersecting  lines.  If  two  lines,  which  do  not  lie  in  a  profile 
plane,  intersect  at  a  point  in  space,  the  horizontal  projections 
of  the  lines  will  intersect  in  the  horizontal  projection  of  the 
point  and  the  vertical  projections  of  the  lines  will  intersect  in 
the  vertical  projection  of  the  point,  Fig.  13.  The  projections  of 
the  common  point  D  must  lie  in  the  same  perpendicular  to  the 
ground  line. 

If  two  lines  do  not  intersect  in  space,  their  projections  will 


14 


DESCRIPTIVE   GEOMETRY 


not  intersect  in  the  same  perpendicular  to  the  ground  line, 
Fig.  14. 

Parallel  lines.     If  two  lines  are  parallel  in  space,  their  pro- 
jections on  the  same  plane  are  parallel,  Fig.  15. 


Pig  12. — Point  C  Fig.  IS.— Intersect-  Fig.  14.— Lines  not  Fig.  15.—Par- 
on  line  AB.  ing  lines.  intersecting.  allel  lines. 


14.  Notation.  Lines  in  space  are  designated  by  capital  let- 
ters as  AB,  CD,  MN,  etc.  Horizontal  projections  of  lines,  small 
letters  as  dby  cd,  mn,  etc.  Vertical  projections  of  lines,  small 
letters  with  prime  marks  as  a'&',  c'd\  m'n\  etc.  Profile  projec- 
tions of  lines,  small  letters  with  double  prime  marks  as  a"!:)", 
c"d";  m"n\  etc. 

The  planes  of  projection  will  be  considered  transparent. 

15.  Problems.  Picture  to  yourself  lines  in  the  following  posi- 
tions and  then  draw  their  projections.  Show  the  profile  pro- 
jections of  lines  in  the  first  and  third  quadrants  in  addition  to 
their  horizontal  and  vertical  projections. 

1.  In  the  first  quadrant,  parallel  to  H  and  oblique  to  V. 

2.  In  the  second  quadrant,  parallel  to  V  and  oblique  to  H. 

3.  In  the  back  of  H  and  inclined  to  V. 

4.  In  the  third  quadrant,  perpendicular  to  H. 

5.  In  the  third  quadrant,  oblique  to  H  and  V.     Draw  the  projections 
of  a  point  on  this  line. 

6.  In  the  first  quadrant  oblique  to  H  and  V  and  in  a  plane  perpen- 
dicular to  G.  L. 


LINES  15 

7.  In  a  plane  bisecting  the  faurth  quadrant. 

8.  An  oblique  line  intersecting  the  ground  line. 

9.  Two  intersecting  lines  in  the  third  quadrant. 

10.  Two  lines  which  are  oblique  to  H  and  V  and  intersect  on  H. 

11.  Given  the  projections  of  a  point  and  the  projections  of  an  oblique 
line.  Draw  the  projections  of  a  line  which  passes  through  the  given 
point  and  is  parallel  to  the  given  line. 

12.  Two  oblique  lines  which  are  parallel,  one  in  the  first  and  the 
other  in  the  second  quadrant. 

13.  Given  the  projections  of  a  point  and  the  projections  of  an  oblique 
line.  Draw  the  projections  of  a  line  which  passes  through  the  given 
point  and  intersects  the  given  line. 

14.  Two  intersecting  lines  in  the  first  quadrant,  one  parallel  to  H  and 
oblique  to  V  and  the  other  parallel  to  V  and  oblique  to  H. 

15.  Two  parallel  lines  which  lie  in  a  profile  plane  and  are  in  the 
third  quadrant. 

16.  Two  intersecting  lines  which  lie  in  a  profile  plane  and  are  in  the 
first  quadrant. 

17.  Two  intersecting  lines  in  the  third  quadrant,  one  perpendicular 
to  H  and  the  other  perpendicular  to  V. 

18.  A  line  is  given  by  its  projections.  Find  the  projections  of  a 
point  of  this  line  which  is  equally  distant  from  H  and  V. 

19.  A  line  of  profile  is  given  by  the  projections  of  two  of  its  points. 
Choose  a  point  of  this  line  which  is  equally  distant  from  H  and  V. 

20.  A  point  and  a  line  lie  in  a  profile  plane.  Determine  by  an  end 
view  whether  or  not  the  point  lies  on  the  line. 

21.  Draw  the  projections  of  a  parallelogram  when  the  projections  of 
two  of  its  adjacent  sides  are  given. 


16  DESCRIPTIVE   GEOMETRY 

PLANES 

16.  In  space,  a  plane  is  fixed  by  three  points,  not  in  the  same 
straight  line,  a  point  and  a  line,  two  intersecting  lines  or  two 
parallel  lines.  If  the  horizontal  and  vertical  projections  of 
these  magnitudes  are  given,  the  plane  which  they  determine  in 
space  will  be  definitely  located.  When  the  statement  ''A  given 
plane"  is  made,  the  student  is  expected  to  draw  the  projections 
of  the  magnitudes  which  represent  the  plane. 

jir^b  The  points  A,  B,  and  C,  Fig.  16,  fix 

^  ^^'^^  \  \  a  plane  in  space.     It  is  evident  that  a 

straight  line  joining  A  and  B,  B  and 


•-^1 


r'"""-^*>.^c  C,  or  A  and  C  will  lie  in  this  plane. 


/< 


If  a  point  and  line  are  given  by 
their    projections,    they    will    fix    a 
y^  ^S^^^ '       .plane.     Other  straight  lines  of  the 

<7>i^ — ^""^  plane  can  be  found  by  joining  the 

Fig.  16.— Points  A,  B,  C  given  point  with  points  on  the  given 

fix  a  plane.  Hj^q 

If  the  plane  is  fixed  by  two  straight  lines,  either  parallel  or 
intersecting,  any  number  of  other  straight  lines  of  the  plane  can 
be  found  by  joining  a  point  in  one  of  the  given  lines  with  a 
point  in  the  other. 

If  a  plane  is  perpendicular  to  the  horizontal  plane  of  projec- 
tion, all  straight  lines  of  the  plane  which  are  of  indefinite 
length,  have  the  same  horizontal  projection.  The  vertical  pro- 
jections of  these  lines  will  be  parallel  or  intersecting  lines. 

If  a  plane  is  perpendicular  to  the  vertical  plane  of  projection, 
all  straight  lines  of  the  plane  which  are  of  indefinite  length 
have  the  same  vertical  projection.  The  horizontal  projections 
of  these  lines  will  be  parallel  or  intersecting  lines. 

If  a  plane  is  perpendicular  to  both  the  horizontal  and  ver- 
tical planes  of  projection,  all  straight  lines  of  the  plane  which 
are  of  indefinite  length  have  the  same  horizontal  projection  and 
also  the  same  vertical  projection. 


PLANES 


17 


If  one  straight  line  of  a  plane  is  parallel  to  the  ground  line, 
the  plane  is  parallel  to  the  ground  line. 

17.  Lines  in  planes.  The  two  intersecting  lines  MN  and  OP, 
Fig.  17,  fix  a  plane.  The  straight  line  AB,  joining  the  point  A 
in  MN  with  the  point  B  in  OPj  is  another  line  of  the  plane,  ah 
is  its  horizontal  and  a'h'  its  vertical  projection. 

A  line  which  lies  in  a  given  plane  and  is  parallel  to  H  is 
called  a  horizontal  of  that  plane.  In  Fig.  18,  HH  is  such  a  line 
which  lies  in  the  plane  of  the  lines  MN  and  OP.  Its  vertical 
projection  Wh'  is  drawn  first  and  then  its  horizontal  projection 


Fig.  17. — AB  line  in  plane  of 
MN  and  OP. 


Fig.  18. — HH  horizontal  of  given 
plane.  FF  frontal  of  given  plane. 


hh  is  found  by  finding  the  horizontal  projections  of  the  points 
B  and  A  where  it  crosses  the  given  lines  MN  and  OP.  A  line 
which  lies  in  a  given  plane  and  is  parallel  to  V,  is  called  a  fron- 
tal of  that  plane.  FF  is  such  a  line  which  lies  in  the  plane  of 
MN  and  OP.  Its  horizontal  projection  //  is  drawn  first  and 
then  its  vertical  projection  ff  is  found.  These  lines  which  lie 
in  a  plane  and  are  parallel  to  H  or  Y  play  an  important  part  in 
the  solution  of  many  problems. 

18.  Points  in  planes.  If  one  projection  of  a  point  which  lies 
in  a  plane  is  given,  the  other  projection  is  found  by  finding  the 
projections  of  a  line  of  the  plane  passing  through  the  point  and 


18  DESCRIPTIVB   GEOMETRY 

then  locating  the  other  projection  of  the  point  on  the  other  pro- 
jection of  the  line. 

All  the  points  which  lie  on  a  given  plane  and  are  a  given  dis- 
tance from  the  horizontal  plane  of  projection  are  on  a  horizon- 
tal of  the  given  plane.  Likewise,  all  the  points  which  lie  on  a 
given  plane  and  are  a  given  distance  from  the  vertical  plane  of 
projection  are  on  a  frontal  of  the  given  plane. 

19.  Problems.  In  the  following  problems  the  given  planes 
may  be  represented  by  the  projections  of  any  two  of  their  lines 
unless  it  is  otherwise  stated. 

1.  Having  given  the  horizontal  projection  of  a  line  which  lies  in  a 
given  plane,  to  find  its  vertical  projection. 

2.  Having  given  the  vertical  projection  of  a  line  which  lies  in  a 
given  plane,  to  find  its  horizontal  projection. 

3.  Having  given  the  horizontal  projection  of  a  point  which  lies  in  a 
given  plane,  to  find  its  vertical  projection. 

4.  Find  the  projections  of  the  locus  of  all  points  which  lie  on  a  given 
oblique  plane  and  are  1"  from  H; 

6.  Find  the  projections  of  the  locus  of  all  points  which  lie  on  a  given 
oblique  plane  and  are  li"  from  V. 

6.  Find  the  projections  of  a  point  which  lies  on  a  given  oblique  plane 
and  is  I''  from  H  and  1"  from  V. 

7.  A  point  and  two  parallel  lines  are  given  by  their  projections.  Is 
the  point  in  the  plane  of  the  two  parallel  lines? 

8.  Having  given  a  plane  which  is  parallel  to  the  ground  line  and  the 
horizontal  projection  of  a  point  of  this  plane.  Find  the  vertical  pro- 
jection of  the  point. 

9.  Having  given  a  plane  which  is  parallel  to  the  ground  line  and  the 
horizontal  projection  of  a  line  of  this  plane.  Find  the  vertical  projec- 
tion of  the  line. 

10.  Having  given  a  plane  which  is  parallel  to  the  ground  line  and  the 
vertical  projection  of  a  line  of  this  plane  which  is  also  parallel  to  the 
ground  line.    Find  the  horizontal  projection  of  the  line. 

11.  Two  lines  in  the  third  quadrant  are  in  such  a  position  that  neither 
their  horizontal  nor  vertical  projections  intersect  within  the  limits  of 
the  drawing.     Determine  graphically  whether  or  not  the  lines  intersect. 

12.  A  plane  is  represented  by  two  intersecting  lines.  The  horizontal 
projection  of  a  line  which  passes  through  their  point  of  intersection 
and  lies  in  the  given  plane  is  given.    Find  its  vertical  projection. 

13.  A  plane  is  represented  by  two  parallel  lines.  The  horizontal  pro- 
jection of  another  line  of  this  plane  which  is  parallel  to  the  first  two 
lines  is  given.    Find  its  vertical  projection. 


REVOLUTION   AND    COUNTER  REVOLUTION  19 

REVOLUTION  AND  COUNTER  REVOLUTION  OF  OBJECTS 

20.  An  object  is  said  to  revolve  about  a  straight  line  as  an 
axis  when  each  of  its  points  moves  in  the  circumference  of  a 
circle  whose  center  is  in  the  axis  and  whose  plane  is  perpen- 
dicular to  the  axis. 

When  an  object  is  revolved  about  a  straight  line  as  an  axis, 
the  relative  position  of  its  points  is  not  changed.  The  object 
can  thus  be  brought  into  a  simpler  position  with  reference  to 
the  planes  of  projection.  The  projections  of  the  object  in  this 
position  are  easily  found  and  from  these  projections  the  projec- 
tions of  the  object  in  its  original  position  are  found  by  the 
counter  revolution  of  its  points. 

REVOLUTION  OF  A  POINT  ABOUT  AN  AXIS 

21.  A  line  perpendicular  to  H  and  a  point  are  given  by  their 
projections.  Revolve  the  point  through  an  angle  of  oc  °  about 
the  line  as  an  axis. 

In  Fig.  19,  let  MN  be  the  axis  and  P  the  given  point. 

The  point  P  will  move  in  the  circumference  of  a  circle  whose 
center  is  at  0  and  whose  radius  is  OP  (Art.  20).  The  plane  of 
the  path  of  this  point  is  perpendicular  to  MN  and  therefore 
parallel  to  H.  Therefore  the  horizontal  projection  of  the  path 
is  a  circle  with  a  radius  op  and  the  vertical  projection  is  a 
straight  line  through  p'  parallel  to  the  ground  line.  Since  the 
plane  of  the  circle  is  parallel  to  H,  the  horizontal  projection  of 
the  angle  through, which  the  radius  sweeps  is  equal  to  the  true 
size  of  the  angle.  Therefore  Pi,  Fig.  19,  is  the  required  posi- 
tion of  P. 

22.  A  line  perpendicular  to  V  and  a  point  are  given  by  their 
projections.  Revolve  the  point  through  an  angle  of  oc  °  about 
the  line  as  an  axis. 

In  Fig.  20,  let  MN  be  the  axis  and  P  the  given  point. 
The  point  P  will  move  in  the  circumference  of  a  circle  whose 
center  is  at  0  and  whose  radius  is  OP  (Art.  20).     The  plane  of 


20 


DESCRIPTIVE   GEOMETRY 


the  path  of  this  point  is  parallel  to  V,  and  therefore  the  vertical 
projection  is  a  circle  with  radius  o'p'  and  the  horizontal  pro- 
jection is  a  straight  line  through  p  parallel  to  the  ground  line. 
Since  the  plane  of  the  circle  is  parallel  to  V,  the  vertical  pro- 
jection of  the  angle  through  which  the  radius  sweeps  is  equal  to 
the  true  size  of  the  angle.  Therefore  Pj,  Fig.  20,  is  the  required 
position  of  P. 

n 


rrP(n 

I 


/A  I' 


m 


o'-l 


1 
I 

I 
I 
I 
I 


Fig. 


19. — P  revolved  through  oc" 
about  MN  as  axis. 


Fig.  20. — P  revolved  through  oc' 
about  MN  as  axis. 


23.  A  line  parallel  to  H  and  oblique  to  V  and  a  point  are 
given  by  their  projections.  Revolve  the  point  about  the  line  as 
an  axis  until  the  point  is  on  the  same  level  as  the  axis. 

In  Fig.  21,  let  MN  be  the  axis  and  P  the  given  point. 

The  point  P  will  move  in  the  circumference  of  a  circle  whose 
center  is  at  0  and  whose  radius  is  OP.  The  plane  of  this  circle 
is  perpendicular  to  the  axis  MN  and  therefore  perpendicular  to 
H.  The  horizontal  projection  of  the  circle  is  the  straight  line 
op  which  passes  through  the  horizontal  projection  p  of  the  point 
and  is  perpendicular  to  the  horizontal  projection  mn  of  the 
axis.  The  vertical  projection  of  the  circle  is  an  ellipse.  To 
find  the  radius  of  the  circle,  let  the  plane  of  the  circle  be  turned 
over  until  it  is  parallel  to  H  about  a  horizontal  axis  which 


REVOLUTION   AND   COUNTER  REVOLUTION 


21 


passes  through  the  center  0  and  is  perpendicular  to  MN.  When 
the  plane  is  in  the  revolved  position,  the  point  P  is  at  pi,  the 
distance  pp^  being  equal  to  the  distance  d.  Then  op^  is  the  re- 
quired radius.  The  point  p^  is  then  moved  around  in  this  cir- 
cumference until  it  cuts  the  line  op  extended  at  Pz  and  p^. 


/  i   /y 


Fig.  21. — Point  P  revolved  about 
MN  as  axis. 


fA. ,.-' 

Fig.  22. — Point  P  revolved  about 
MN  as  axis. 


When  the  plane  of  the  circle  is  counter  revolved  to  its  former 
position  at  right  angles  to  MN,  the  points  Pg  and  P3  do  not  move 
and  therefore  remain  on  the  same  level  as  MN.  Then  the  points 
P2  and  P3  are  the  required  positions  of  the  point  P. 

The  perpendicular  distance  from  any  point  in  the  circumfer- 
ence P2P1P3  to  the  line  op  represents  the  distance  of  the  point  P 
above  or  below  MN  when  the  plane  of  the  circle  stands  perpen- 
dicular to  MN. 

24.  A  line  parallel  to  V  and  oblique  to  H  and  a  point  are 
given  by  their  projections.  Revolve  the  point  about  the  line 
as  an  axis  until  the  point  is  the  same  distance  from  V  as  the  axis. 

In  Fig.  22,  let  MN  be  the  axis  and  P  the  given  point. 

The  point  P  will  move  in  the  circumference  of  a  circle  whose 
center  is  at  0  and  whose  radius  is  OP.     The  plane  of  this  circle 


22  DESCRIPTIVE   GEOMETRY 

is  perpendicular  to  the  axis  MN  and  therefore  perpendicular  to 
V.  The  vertical  projection  of  the  circle  is  the  straight  line  o'y' 
which  passes  through  the  vertical  projection  p'  of  the  point  and 
is  perpendicular  to  the  vertical  projection  m'w'  of  the  axis. 
The  horizontal  projection  of  the  circle  is  an  ellipse.  To  find  the 
radius  of  the  circle,  let  the  plane  of  the  circle  be  turned  over 
until  it  is  parallel  to  V  about  an  axis  parallel  to  V  which  passes 
through  the  center  0  and  is  perpendicular  to  MN.  When  the 
plane  is  in  the  revolved  position,  the  point  P  is  at  p'l,  the  dis- 
tance p'p'i  being  equal  to  the  distance  d  Then  o''g\  is  the  re- 
quired radius.  The  point  'p\  is  then  moved  around  in  this  cir- 
cumference until  it  cuts  the  line  dp'  at  y'^  and  p'g.  "When  the 
plane  of  the  circle  is  counter  revolved  to  its  former  position  at 
right  angles  to  MN,  the  points  Pg  and  Pg  do  not  move  and  there- 
fore remain  the  same  distance  from  Y  as  MN.  Then  the  points 
P2  and  P3  are  the  required  positions  of  the  point  P. 

The  perpendicular  distance  from  any  point  in  the  circumfer- 
ence p'zP'xP'z  to  the  line  o'p'  represents  the  distance  of  the  point 
P  in  front  of  or  behind  MN  when  the  plane  of  the  circle  stands 
perpendicular  to  MN. 

25.  Problems. 

Note. — Unless  stated  otherwise  all  problems  are  to  be  solved  in  the 
third  quadrant. 

1.  Revolve  a  point  which  is  in  the  first  quadrant  through  an  angle  of 
45  degrees  about  an  axis  perpendicular  to  H  and  show  its  projections  In 
the  new  position. 

2.  Revolve  a  point  which  is  in  the  fourth  quadrant  into  H  about  an 
axis  in  H. 

3.  Revolve  a  point  which  is  in  the  third  quadrant  into  H  about  the 
ground  line  as  an  axis. 

4.  Having  given  a  point  in  H  and  an  axis  in  H  30°  to  V,  revolve  the 
point  about  the  axis  until  it  is  1"  above  H  and  show  its  projections  in 
this  position. 

5.  Given  a  point  in  H  also  a  line  in  H  oblique  to  V.  Revolve  the 
point  about  the  line  as  an  axis  until  the  point  strikes  V.  Show  its 
projections  in  this  position. 

6.  Given  a  point  in  V  and  a  line  in  V  oblique  to  H.  Revolve  the  point 
about  the  line  as  an  axis  until  the  point  is  1"  behind  V.  Show  its 
projections  in  this  position. 


'REVOLUTION  AND   COUNTER  REVOLUTION  23 

7.  Given  a  point  in  V  and  a  line  in  H  oblique  to  V.  Revolve  the  point 
into  H  about  the  line  as  an  axis  and  show  its  projections  in  this  posi- 
tion. 

8.  A  line  lies  in  the  front  part  of  H  and  is  oblique  to  the  ground  line. 
A  point  lies  in  the  back  part  of  H.  Revolve  the  point  about  the  line  as 
an  axis  until  it  strikes  V.     Show  its  projections  in  this  position. 

9.  Given  an  axis  which  is  oblique  to  V  and  is  parallel  to  and  i"  above 
H.  Revolve  a  point  which  is  in  the  first  quadrant  through  an  angle  of 
90°  about  this  axis.     Show  its  projections  in  this  position. 

10.  Given  an  axis  which  is  oblique  to  V  and  is  parallel  to  and  i" 
below  H.  Revolve  a  point  about  this  axis  until  the  point  is  V  below 
H.    Show  its  projections  in  this  position. 

11.  Given  an  axis  which  is  oblique  to  V  and  is  parallel  to  and  f" 
above  H.  Revolve  a  point  which  is  in  the  first  quadrant  about  the 
axis  until  it  is  in  the  second  quadrant.  Show  its  projections  in  this 
position. 

12.  Given  a  point  in  the  third  quadrant  and  an  axis  in  H  oblique  to  V. 
Revolve  the  point  about  the  axis  and  find  the  projections  of  the  points 
where  it  passes  through  H  and  V. 

13.  Given  a  point  which  is  in  V,  and  a  line  in  H.  Revolve  the  point 
about  the  line  as  an  axis  until  the  point  is  1"  in  front  of  V.  Show  its 
projections  in  this  position. 

14.  Revolve  a  point  A  which  is  2"  above  H  and  is  li"  in  front  of  V 
about  an  axis  MN  which  is  in  H  and  makes  30°  with  V  until  the  point 
is  i"  behind  V  and  is  below  H.     Show  its  projections  in  this  position. 

15.  Given  two  points  A  and  B  on  H  and  unequal  distances  from  V. 
Find  a  point  on  V  which  is  2''  from  both  A  and  B. 

16.  Two  lines  lie  on  H,  oblique  to  V  and  made  60°  with  each  other. 
Revolve  the  one  about  the  other  as  an  axis  until  the  H  projection  of 
the  angle  between  them  is  30°.     Show  their  projections  in  this  position. 

17.  Given  two  intersecting  lines  in  H.  Revolve  the  one  about  the 
other  until  a  point  of  the  moving  line  is  in  V.  Show  the  projections 
of  the  lines  in  this  position. 

18.  Two  parallel  lines  are  in  H  oblique  to  V.  Revolve  one  line  about 
the  other  as  an  axis  until  a  point  of  the  moving  line  strikes  V.  Show 
their  projections  in  this  position. 

19.  Two  intersecting  lines  are  in  H,  oblique  to  V.  Revolve  one 
about  the  other  until  a  point  of  the  moving  line  is  1"  below  H.  Show 
their  projections  in  this  position. 

20.  Given  an  equilateral  triangle  on  H.  Revolve  the  triangle  about 
one  side  as  an  axis  until  the  H  projection  of  the  angle  opposite  the 
axis  is  a  right  angle.  Show  the  projections  of  the  triangle  in  this 
position. 


24  DESCRIPTIVE  GEOMETRY 

21.  Given  a  right  triangle  in  H.  Revolve  the  triangle  about  the 
hypothenuse  as  an  axis  until  the  H  projection  of  the  right  angle  is  an 
angle  of  120°.     Show  the  projections  of  the  triangle  in  this  position. 

22.  Given  an  isosceles  triangle  in  H  oblique  to  V.  The  angle  at  the 
vertex  is  30°.  Revolve  the  triangle  about  the  base  as  an  axis  until  the 
H  projection  of  the  30°  angle  is  a  right  angle.  Shov?"  the  projectione 
of  the  triangle  in  this  position. 

23.  Given  a  scalene  triangle  on  H  oblique  to  V  (all  angles  acute). 
Revolve  the  smallest  angle  about  the  side  opposite  as  an  axis  until  the 
H  projection  of  the  moving  angle  is  a  right  angle.  Show  the  projec- 
tions of  the  triangle  in  this  position. 

24.  Revolve  a  line  AB  which  is  in  H  about  a  line  CD,  also  in  H,  until 
AB  makes  an  angle  of  30°  with  H.  Show  the  projections  of  AB  in  this 
position. 

25.  Given  an  axis  on  H  oblique  to  V  and  a  point  on  H.  Revolve  the 
point  through  an  angle  of  30°  about  the  axis  and  show  its  projections 
in  this  position. 

26.  Given  an  axis  on  H  oblique  to  V  and  a  li"  equilateral  triangle  on 
H.  Revolve  the  triangle  through  an  angle  of  45°  about  the  axis  and 
show  its  projections  in  this  position. 

27.  Given  an  axis  on  H  and  a  1^"  square  on  H  with  sides  oblique  to 
the  axis.  Revolve  the  square  through  an  angle  of  60°  about  the  axis 
and  show  its  projections  in  this  position. 

LINE  CONVENTIONS 

26.  Pencil.  The  planes  of  projection  are  considered  trans- 
parent. 

Retrace  required  lines  so  that  they  stand  out  from  the  con- 
struction making  hidden  lines  dashed.     The  dashes  should  be 
about  1/8"  long  and  1/32"  apart. 
/AH  other  construction  is  to  be  in  very  fine  full  lines. 

Ink.  Given  lines  when  visible,  fine,  full,  black  lines;  when 
invisible,  fine,  dashed,  black  lines. 

Required  lines  when  visible,  heavy,  full,  black  lines;  when  in- 
visible, heavy,  dashed,  black  lines. 

All  other  construction  is  to  be  in  fine,  full,  red  lines. 


CHAPTER  II 

THE   ELEMENTARY  PRINCIPLES   OF   THE   POINT, 
STRAIGHT  LINE,  AND  PLANE. 


27.  The  horizontal  and  vertical  projections  of  a  point  will  fix 
a  definite  point  in  space  if  the  ground  line  is  shown  on  the 
drawing  (Art.  7).  The  distances  from  the  projections  of  the 
point  to  the  ground  line  are  equal  respectively  to  the  distances 
of  the  point  from  the  planes  of  projection.  If  the  ground  line 
is  not  shown,  the  projections  of  a  point  will  not  fix  a  definite 
point  in  space,  since  there  is  nothing  to  show  its  distances  from 
the  planes  of  projection.  If,  however,  the  projections  of  two 
or  more  points  are  given  without  the  ground  line  being  shown, 
the  vertical  projections  will  show  the  relative  heights  of  the 
points  and  the  horizontal  projections  will  show  their  relative 
distances  from  the  vertical  plane. 

The  projection  of  an  object  on  a  plane  does  not  depend  upon 
its  distance  from  the  plane,  but  upon  the  relative  distances  of 
its  points  from  that  plane.  For  example,  the  horizontal  pro- 
jection of  a  cube  does  not  depend  upon  the  distance  of  the  cube 
from  the  horizontal  plane,  for  the  horizontal  projection  will  not 
be  changed  if  all  the  corners  of  the  cube  be  moved  the  same 
distance  up  or  down  along  vertical  lines.  The  horizontal  pro- 
jection of  the  cube  will  be  changed,  however,  if  one  or  two 
comers  remain  fixed  while  the  other  corners  are  moved.  Two 
projections  of  an  object,  when  the  ground  line  is  omitted,  will 
in  general  definitely  determine  the  form  of  the  object,  but  will 
not  show  its  distances  from  the  planes  of  projection. 

Hereafter  the  ground  line  will  be  omitted  from  many  of  the 
drawings,  but  it  is  always  understood  to  be  at  right  angles  to 
the  line  joining  two  projections  of  the  same  point. 


26 


DESCRIPTIVE  GEOMETRY 


28.  A  straight  line  is  given  by  its  projections.    Find  the  true 
length  of  the  line  and  the  angles  which  it  makes  with  H  and  V. 

Let  AB,  Fig.  23,  be  the  given  line. 

Analysis.     Revolve  the  horizontal  projecting  plane  of  the  line 
about  an  axis  which  cuts  the  line,  and  is  parallel  to  its  horizon- 


FiG.  23. — a&^  and  a'b'^  true  length  of  AB 
och  and  ocv  angles  with  H  and  V. 


tal  projection,  until  the  plane  is  parallel  to  H.  The  horizontal 
projection  of  the  line  in  this  position  is  the  true  length  of  the 
line. 

The  angle  between  the  line  in  its  revolved  position  and  the 
axis  is  equal  to  the  angle  which  the  line  makes  with  H.  The 
horizontal  projection  of  the  angle  in  this  position  is  the  true 
size  of  the  angle. 

Construction.  Let  ac  and  a'&  be  the  projections  of  the  axis" 
which  passes  through  A  and  is  parallel  to  ah.    After  AB  is  re- 


PROBLEMS  IN  POINT,   STRAIGHT  LINE,   AND  PLANE  27 

volved  about  AC  as  an  axis,  the  horizontal  projection  of  AB  is 
a&i.  &&1  is  equal  to  h'c'.  The  point  A,  being  on  the  axis,  is  sta- 
tionary. Then  h^a  is  the  true  length  of  the  line  and  the  angle 
h^ab  is  the  true  size  of  the  angle  which  AB  makes  with  H. 

To  find  the  angle  which  the  line  AB  makes  with  V,  revolve 
the  vertical  projecting  plane  of  the  line  about  an  axis  AE  which 
passes  through  A  and  is  parallel  to  the  vertical  projection  of 
AB.  When  this  projecting  plane  is  parallel  with  Y,  the  ver- 
tical projection  h'^a'!}'  is  the  true  size  of  the  angle  which  the  line 
AB  makes  with  V.  a'h'2  is  the  true  length  of  AB  and  is  there- 
fore equal  to  ab-^^. 

29.  Conversely,  given  the  horizontal  (or  vertical)  projection 
of  a  line  and  the  angle  which  it  makes  with  H  (or  V)  to  find  the 
other  projections  of  the  line. 

The  student  should  make  the  construction  for  this  problem. 

Note.  In  Fig.  23,  prove  that  the  angle  h-^ac  is  or  is  not  equal 
to  the  angle  h'a'c'.  Also  prove  that  the  angle  h^a'e'  is  or  is  not 
equal  to  the  'angle  hae. 

30.  Problems. 

1.  Find  the  length  of  a  straight  line  joining  any  two  points  in  a 
profile  plane.    Find  the  angles  which  this  line  makes  with  H  and  V. 

2.  From  a  point  on  a  line,  given  by  its  projections,  lay  off  along  the 
line  a  length  of  two  units. 

3.  A  square  pyramid  has  its  base  in  a  horizontal  plane.  Find  the 
true  length  of  the  slant  height  and  a  lateral  edge  of  the  pyramid. 

4.  The  horizontal  projection  of  a  line  is  two  units  long  and  makes 
45°  with  V.    If  the  line  makes  30°  with  H,  find  its  vertical  projection. 

5.  The  horizontal  projection  of  a  line  is  three  units  long  and  makes 
30°  with  V.     If  the  line  is  four  units  long,  find  its  vertical  projection. 

6.  A  line  is  four  units  long  and  its  horizontal  projection  makes  15° 
with  V.  Find  its  vertical  projection  when  one  end  of  the  line  is  two 
units  higher  than  the  other. 

7.  Draw  the  projections  of  a  line  which  makes  60°  with  H  and  is 
oblique  to  V. 

8.  Draw  the  projections  of  a  line  which  makes  45°  with  V  and  is 
oblique  to  H. 

9.  Draw  the  projections  of  a  line  which  is  four  units  long,  makes  45° 
with  H  and  is  oblique  to  V. 


28  DESCRIPTIVE   GEOMETRY 

10.  Draw  the  projections  of  a  line  which  is  four  units  long,  makes 
30°  with  V  and  is  oblique  to  H. 

11.  Draw  the  projections  of  a  line  which  makes  30"  with  H,  is  oblique 
to  V  and  has  one  end  three  units  higher  than  the  other. 

12.  Draw  the  projections  of  a  line  which  makes  60°  with  V,  is  oblique 
to  H  and  has  one  end  four  units  in  front  of  the  other. 

13.  Draw  the  projections  of  a  line  which  lies  in  a  profile  plane,  makes^ 
60°  with  H  and  is  four  units  long.  What  angle  does  this  line  make 
with  V? 

14.  A  li"  cube  has  its  base  parallel  to  H  and  a  side  face  30°  to  V. 
Find  the  true  length  of  a  diagonal  of  the  cube  and  the  angle  which  it 
makes  with  an  edge  of  the  cube. 

15.  Find  the  true  length  of  the  hip  rafter  MN  in  the  roof  shown  in 
Fig.  37,  page  49.    Also  find  the  true  length  of  the  jack  rafter  OQ. 

16.  Find  the  true  length  of  the  edge  MN  in  the  ventilator  pipe  shown 
in  Fig.  36,  page  49. 

17.  Find  the  true  length  of  the  edge  MN  in  the  chute  shown  in 
Fig.  38,  page  49. 

31.  Two  parallel  lines  are  given  by  their  projections.  Find 
the  angles  which  the  plane  of  these  lines  makes  with  the  planes 
of  projection. 

Let  AB  and  CE,  Fig.  24,  be  the  given  lines. 

Analysis.  A  line  of  the  given  plane  vrhich  is  perpendicular 
to  a  horizontal  will  make  the  same  angle  with  H  as  the  given 
plane.  A  line  of  the  given  plane  which  is  perpendicular  to  a 
frontal  will  make  the  same  angle  with  V  as  the  given  plane. 
Therefore,  draw  these  lines  and  find  the  angles  which  they 
make  with  H  and  V  respectively.  These  are  the  required  angles. 

Construction,  h'h'  and  Kh  are  the  projections  of  a  horizontal 
on  the  given  plane  (Art.  17).  A  line  which  is  perpendicular  to 
a  horizontal  has  its  horizontal  projection  perpendicular  to  the 
horizontal  projection  of  the  horizontal.  In  like  manner  a  line 
which  is  perpendicular  to  a  frontal  has  its  vertical  projection 
perpendicular  to  the  vertical  projection  of  the  frontal.  There- 
fore draw  ge  perpendicular  to  hh.  Then  ge  and  g'e'  are  the  pro- 
jections of  a  line  which  is  perpendicular  to  HH  and  lies  on  the 
given  plane,     oc  ^  is  the  angle  which  the  line  GE  makes  with  H 


PROBLEMS  IN  POINT,   STRAIGHT  LINE,   AND  PLANE 


29 


(Art.  28),  and  is  therefore  the  angle  which  the  plane  of  the 
given  lines  makes  with  H. 

By  taking  a  line  of  the  plane  which  is  perpendicular  to  the 
frontal  FF,  the  angle  oc  ^  which  the  plane  makes  with  V  is  found. 


Fig.  24. — och  angle  plane  makes  with  H. 
ocy  angle  plane  makes  with  V. 


32.  Conversely,  to  draw  the  projections  of  two  lines,  the  plane 
of  which  makes  a  given  angle  with  H  or  V. 

Analysis.  Draw  the  projections  of  a  horizontal  which  is  ob- 
lique to  V.  Draw  the  projections  of  a  line  which  is  at  right 
angles  to  this  horizontal  and  makes  the  required  angle  with  H 
(Art.  29).  These  are  the  required  lines,  the  plane  of  which 
makes  the  required  angle  with  H. 

Fig.  25  shows  the  construction  for  a  plane  which  makes  the 
angle  oc  with  H ;  HH  and  AB  being  the  required  lines. 

By  using  a  frontal  and  a  line  at  right  angles  to  it  making  the 
given  angle  with  V,  a  plane  is  fixed  which  makes  the  required 
angle  with  Y,  Fig.  26. 


30 


DESCRIPTIVE   GEOMETRY 


Other  lines  on  these  planes  can  be  drawn  by  joining  points  in 
one  of  the  lines  with  points  in  the  other  line. 


Fig.  2^.— Plane  oz°  with  H. 


Fig.  2%.— Plane  oc°  with  y. 


33.  Problems. 

1.  Find  the  angles  which  the  plane  of  two  intersecting  lines  makes 
with  H  and  V. 

2.  Given  two  intersecting  lines,  one  parallel  to  G.  L.  and  the  other 
oblique  to  H  and  V,  find  the  angles  which  the  plane  of  the  lines  makes 
with  H  and  V.    What  is  the  relation  between  the  angles? 

3.  Through  a  given  point  draw  two  lines  the  plane  which  makes  30° 
with  H.     Does  this  plane  make  60°  with  V? 

4.  Given  two  intersecting  lines,  one  oblique  to  G.  L.  and  the  other  a 
line  of  profile,  find  the  angles  which  the  plane  of  the  lines  makes  with 
H  and  V. 

5.  Draw  the  projections  of  (a)  an  equilateral  triangle,  (b)  square, 
(c)  regular  hexagon,  when  the  plane  of  the  triangle  makes  (a)  30°, 
(b)  45°,  (c)  60°  with  H  and  is  oblique  to  V. 

6.  Two  intersecting  lines  are  given  by  their  projections.  Draw  the 
projections  of  an  equilateral  triangle  lying  in  the  plane  of  the  two  lines 
so  that  one  side  of  the  triangle  makes  30°  with  a  horizontal  of  the  plane. 

7.  Two  parallel  lines  of  profile  lie  in  separate  profile  planes.  The 
plane  of  the  lines  is  to  be  oblique  to  the  ground  line.  Find  the  angles 
which  this  plane  makes  with  H  and  V. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,    AND   PLANE  31 

8.  A  line  AB  makes  45"  with  H  and  is  oblique  to  V.  Represent  a 
plane  which  contains  the  line  and  mak:es  60°  with  H.  (Use  a  right 
circular  cone  with  vertex  on  AB,  base  parallel  to  H  and  elements  60" 
with  H). 

34.  If  an  oblique  plane  is  given  by  two  intersecting  straight 
lines,  the  angles  between  the  projections  of  these  lines  are  not 
usually  the  true  size  of  the  angles  between  the  lines.  If  the 
plane  of  the  lines  is  revolved  until  it  is  parallel  to  one  of  the 
planes  of  projection,  then  the  angles  between  the  projections  of 
the  lines  on  that  plane  are  the  true  size  of  the  angles  between 
the  lines. 

If  the  projections  of  a  plane  figure  such  as  a  square  or  a 
triangle  are  given,  it  is  necessary  to  revolve  the  plane  of  the 
figure  until  it  is  parallel  to  one  of  the  planes  of  projection  be- 
fore its  projection  on  that  plane  is  the  true  size  of  the  figure. 
Conversely,  to  find  the  projections  of  a  figure  which  lies  on  a 
given  plane,  it  is  necessary  to  revolve  the  plane  until  it  is  par- 
allel to  one  of  the  planes  of  projection.  While  the  plane  is  in 
this  position,  construct  the  true  size  of  the  figure  on  it.  The 
plane  and  the  figure  must  then  be  counter  revolved  to  the  origi- 
nal position  of  the  plane  and  the  projections  of  the  figure  in  this 
position  found. 

35.  Two  intersecting  lines  axe  given  by  their  projections. 
Find  the  true  size  of  the  angle  between  them. 

Let  AO  and  BO,  Fig.  27,  be  the  given  lines. 

Analysis.  Draw  a  horizontal  in  the  plane  of  the  given  lines. 
Revolve  the  given  lines  about  the  horizontal  as  an  axis  until  the 
plane  of  the  lines  is  parallel  to  H.  The  horizontal  projection 
of  the  angle  in  this  position  is  the  true  size  of  the  angle  between 
the  lines  since  they  do  not  change  their  relative  position  dur- 
ing the  revolution. 

Construction,  h'h'  is  the  vertical  and  lih  the  horizontal  pro- 
jection of  a  horizontal  in  the  plane  of  AO  and  BO  (Art.  17). 
When  the  given  lines  are  revolved  into  a  horizontal  position 
about  HH  as  an  axis,  their  point  of  intersection  0  moves  to  Og 
(Art.  23).     The  points  X  and  Y  on  the  axis  do  not  move.     Then 


32 


DESCRIPTIVE  GEOMETRY 


xOzV  is  the  true  size  of  the  angle  between  the  lines  AO  and  BO. 

The  problem  can  be  solved  by  revolving  the  plane  of  the 
angle  until  it  is  parallel  to  V  about  a  frontal  as  an  axis. 

36.  To  bisect  an  angle.  When  the  angle  is  shown  in  its  true 
size,  the  bisector  can  be  drawn.  The  bisector  can  then  be  re- 
volved back  until  the  plane  of  the  angle  is  in  its  original  posi- 


FiG.  27. — xo^y  angle  tetween  AO  and  BO. 

tion  and  the  projections  of  the  bisector  found.  In  general,  the 
projections  of  the  bisector  will  not  bisect  the  projections  of  the 
angle. 

The  projection  of  an  angle  can  be  larger  than,  equal  to,  or 
smaller  than  the  angle  itself.  The  projection  of  a  right  angle 
is  a  right  angle  when  one  side  of  the  angle  is  parallel  to  the 
plane  upon  which  the  projection  is  made. 

37.  Problems. 

1.  Find  the  angle  between  two  intersecting  lines  by  using  a  frontal 
as  an  axis. 

2.  Find  the  angle  between  two  intersecting  lines  when  one  is  oblique 
to  H  and  V  and  the  other  is  parallel  to  H  and  makes  30°  to  V. 

3.  Find  the  angle  between  two  intersecting  lines  when  one  is  oblique 
to  H  and  V  and  the  other  is  parallel  to  the  ground  line. 


PROBLEMS  IN  POINT,   STRAIGHT  LINE,   AND  PLANE  33 

4.  Draw  the  projections  of  the  bisectors  of  the  angles  of  a  given  tri- 
angle. 

5.  Draw  the  projections  of  a  triangular  pyramid  with  base  parallel 
to  H.    Find  the  true  size  of  the  angle  between  any  two  lateral  edges. 

6.  Given  two  intersecting  lines  AB  and  CD,  AB  parallel  to  H  and  45'' 
to  V  and  CD  parallel  to  V  and  30°  to  H,  draw  the  projections  of  the 
bisector  of  the  angle  between  them. 

7.  A  triangle  lies  in  a  plane  which  is  perpendicular  to  V  and  30° 
to  H.    Find  the  true  size  of  the  triangle  by  using  a  frontal  as  an  axis. 

8.  The  plane  of  two  intersecting  lines  is  perpendicular  to  H  and  is 
oblique  to  V.  Find  the  true  size  of  the  angle  between  the  lines  by  using 
a  frontal  as  an  axis. 

9.  Find  the  angle  between  two  intersecting  lines  when  one  is  oblique 
to  H  and  V  and  one  lies  in  a  profile  plane. 

10.  Given  both  projections  of  three  corners  of  a  quadrilateral  and  the 
horizontal  projection  of  the  fourth  comer,  draw  the  projections  of  the. 
quadrilateral  and  find  its  true  size  and  shape. 

11.  Letter  two  points  on  the  projections  of  the  bisector  of  the  angle 
between  two  lines  which  lie  in  a  profile  plane. 

12.  Find  the  true  size  of  the  angle  NMP  in  the  chute  shown  in 
Fig.  38,  page  49. 

38.  A  point  and  a  line  are  given  by  their  projections.  Draw 
the  projections  of  a  line  which  passes  through  the  point  and 
makes  a  given  angle  with  the  given  line. 

Let  P  be  the  given  point,  oc  the  given  angle,  and  AB  the  given 
line,  Fig.  28. 

Analysis.  Eevolve  the  plane  of  the  point  and  line  about  a 
horizontal  until  it  is  parallel  to  H.  From  the  revolved  position 
of  the  point,  draw  a  line  which  makes  the  angle  oc  with  the 
revolved  position  of  the  given  line.  When  this  line  is  revolved 
back  with  the  plane  to  the  original  position  of  the  plane,  it  will 
be  the  required  line. 

Construction,  h'h'  is  the  vertical  projection  and  hh  is  the  hor- 
izontal projection  of  a  horizontal  which  lies  on  the  plane  of  P 
and  AB  and  cuts  AB  at  the  point  C  (Art.  17).  When  the  plane 
is  revolved  about  HH  as  an  axis,  B  moves  to  Bg  and  P  and  0 
remain  fixed,  being  on  the  axis.  ftgC  is  the  horizontal  projection 
of  the  revolved  position  of  AB.  Through  p  draw  pe^,  making 
the  required  angle  oc  with  h^c     e  and  e'  are  the  projections  of 


34 


DESCRIPTIVE   GEOMETRY 


the  point  E  after  the  counter  revolution  of  the  plane.  Then  pe 
is  the  horizontal  and  p'e'  is  the  vertical  projection  of  the  re- 
quired line. 


Fig.  2S—PE  oc°  loith  AB. 

39.  Problems. 

1.  Solve  the  above  problem  when   oc  is   (a)   30°,   (b)   45**,   (c)   60°, 
(d)  75^ 

2.  Draw  the  projections  of  a  line  which  is  the  shortest  distance  from 
a  given  point  to  a  given  oblique  line. 

3.  Draw  the  projections  of  a  line  which  is  the  shortest  distance  from 
a  given  point  to  a  given  horizontal  line. 

4.  Draw  the  projections  of  a  line  which  is  the  shortest  distance  from 
a  given  point  to  a  given  line  parallel  to  V. 

5.  Draw  the  projections  of  a  line  which  is  the  shortest  distance  from 
a  given  point  to  a  given  line  of  profile. 

6.  Find  the  projections  of  a  point  which  is  three  units  from  a  given 
point  P  and  lies  on  a  given  line  AB. 

7.  Find  the  distance  between  two  parallel  lines  which  are  oblique  to 
H  and  V. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,    AND   PLANE  35 

8.  A  point  and  an  oblique  line  are  given  by  their  projections.  Draw 
the  projections  of  an  (a)  equilateral  triangle,  (b)  square,  (c)  regular 
hexagon  with  center  at  the  given  point  and  side  along  the  given  line. 

9.  Draw  the  projections  of  a  line  which  passes  through  a  given  point 
P  and  makes  (a)  30°,  (b)  45°,  (c)  60°,  (d)  75°  with  a  given  line  of 
profile. 

10.  A  line  AB  parallel  to  the  ground  line  and  a  point  P  are  given  by 
their  projections.  A  2"  square  lies  on  the  plane  of  the  point  and  line. 
Draw  the  projections  of  the  square  when  one  corner  is  on  AB  and  one 
side  passes  through  P  and  makes  30°  with  AB. 

11.  Find  the  distance  from  the  point  P  to  the  line  MN  in  the  chute 
shown  in  Fig.  38,  page  49. 

40.  A  point  and  two  lines  are  given  by  their  projections. 
Represent  a  plane  which  contains  the  given  point  and  is  parallel 
to  the  given  lines. 

Analysis.  Through  the  given  point  draw  a  line  parallel  to 
each  of  the  given  lines.  The  plane  of  these  two  lines  is  the  re- 
quired plane. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

41.  Problems. 

1.  Represent  a  plane  which  contains  a  given  line  and  is  parallel  to 
another  given  line. 

2.  Represent  a  plane  which  contains  a  given  point  and  is  parallel 
to  G.  L. 

3.  Two  lines  parallel  to  G.  L.  are  given  by  their  projections.  Rep- 
resent a  plane  which  contains  a  given  point  and  is  parallel  to  the 
given  lines. 

4.  Two  lines  are  given  by  their  projections,  one  is  oblique  to  G.  L. 
and  the  other  is  a  line  of  profile.  Represent  a  plane  which  contains  a 
given  point  and  is  parallel  to  the  given  lines. 

5.  Represent  a  plane  which  contains  a  given  point  and  passes  at  equal 
distances  from  two  other  given  points. 

6.  Represent  a  plane  which  contains  a  given  line  and  passes  at  equal 
distances  from  two  given  points. 

7.  Represent  a  plane  which  contains  a  given  point  and  passes  at 
equal  distances  from  three  other  given  points. 


36 


DESCRIPTIVE   GEOMETRY 


42.  A  plane  is  represented  by  two  of  its  lines.  Find  the  point 
in  which  a  given  oblique  line  pierces  this  plane. 

This  problem  should  be  thoroughly  mastered.  It  is  used  in' 
finding  the  plane  sections  of  all  ruled  surfaces,  the  intersections 
of  surfaces  with  plane  faces  and  indirectly  to  find  the  intersec- 
tions of  such  curved  surfaces  as  cylinders  and  cones. 

Let  AB  and  AC,  Fig.  29,  be  the  lines  which  represent  the 
plane  and  MN  the  given  oblique  line. 

jrr  ^b 


Fig.  29. — MN  pierces  plane  of  AB  and  AC  at  P. 

Analysis.  Find  the  points  where  the  lines  which  represent 
the  plane  pierce  the  horizontal  or  vertical  projecting  plane  of 
the  given  oblique  line.  The  line  joining  these  two  points  is  the 
line  of  intersection  of  the  given  plane  with  the  projecting  plane 
of  the  oblique  line.  This  line  of  intersection  cuts  the  given 
line  in  the  required  point. 

Construction.  AB  pierces  the  horizontal  projecting  plane  of 
MN  at  Y  and  AC  pierces  it  at  X.  x'y\  the  vertical  projection 
of  the  line  joining  these  two  points,  intersects  mV  at  p\  the 
vertical  projection  of  the  required  point;  p  is  its  horizontal 
projection. 

43.  Problems. 

1.  Find  where  a  line  pierces  the  plane  of  two  parallel  lines. 

2.  Find  where  a  line  pierces  the  plane  of  a  point  and  a  line. 


PROBLEMS   IN   POINT,    STRAIGHT  LINE,   AND  PLANE  37 

3.  Find  where  a  line  pierces  the  plane  of  three  points. 

4.  Find  the  point  where  a  line  which  is  parallel  to  G.  L.  pierces  a 
.plane  which  is  oblique  to  H  and  V. 

5.  Find  the  point  where  a  line  which  is  parallel  to  H  and  oblique  to 
V  pierces  a  plane  which  is  parallel  to  G.  L, 

6.  rfnd  the  point  where  a  line  which  lies  in  a  profile  plane  pierces  a 
plane  which  is  parallel  to  G.  L. 

7.  Two  planes  are  each  represented  by  two  of  their  lines.  Find  the 
length  of  the  part  of  an  oblique  line  included  between  the  two  planes. 

8.  An  oblique  triangular  pyramid  with  its  base  parallel  to  H  is  given 
by  its  projections.  Find  the  intersection  of  this  pyramid  with  a  given 
oblique  plane. 

9.  Find  the  intersection  of  a  given  oblique  line  with  a  plane  which 
makes  60°  with  H  and  is  oblique  to  V. 

10.  A  line  makes  45°  with  H  and  is  oblique  to  V.  Find  the  intersec- 
tion of  this  line  with  a  given  oblique  plane. 

11.  Draw  the  projections  of  a  line  which  contains  a  given  point  and 
touches  any  other  two  non-intersecting  lines. 

12.  Draw  the  projections  of  a  line  whick  contains  a  given  point,  is 
parallel  to  a  given  plane,  and  touches  another  given  line. 

13.  The  same  as  problem  12  when  the  given  line  is  a  line  of  profile. 

14.  Given  three  lines,  no  two  of  which  lie  in  the  same  plane,  draw  the 
projections  of  a  line  which  touches  two  of  them  and  is  parallel  to  the 
third. 

15.  Given  three  lines,  no  two  of  which  lie  in  the  same  plane,  find  the 
projections  of  a  line  which  touches  all  three' of  them. 

16.  Given  the  projections  of  an  oblique  line  and  a  line  of  profile,  draw 
the  projections  of  a  line  which  touches  the  given  lines  and  is  parallel 
to  the  ground  line. 

44.  Two  planes  are  each  represented  by  two  of  their  lines. 
Find  the  line  of  intersection  of  the  two  planes. 

First  Method.  Let  AB  and  BC,  Fig.  30,  be  the  lines  of  one 
plane  and  MN  and  OP  the  lines  of  the  other  plane. 

Analysis.  Find  the  points  where  the  two  lines  of  one  plane 
pierce  the  other  plane.  The  line  joining  these  two  points  is  the 
required  line  of  intersection  of  the  two  planes. 

Construction.  MN  pierces  the  plane  of  AB  and  BC  at  the 
point  K  (Art.  42).  OP  pierces  the  plane  of  AB  and  BC  at  the 
point  G.  Then  ¥g'  and  kg  are  the  vertical  and  horizontal  pro- 
jections of  the  required  line  of  intersection  of  the  two  planes. 


38 


DESCRIPTIVE   GEOMETRY 


Second  Method.     Let  OA  and  OB,  Fig.  31,  be  the  lines  of  one 
plane  and  CD  and  CE  the  lines  of  the  other  plane. 

.0 


Fig.  30. — KG  line  of  intersection  of  two  planes. 
Analysis.    Any  auxiliary  plane  will  cut  lines  from  the  given 
planes  and  these  lines  will  intersect  in  a  point  which  lies  on  the 


Fig.  31. — KG  line  of  intersection  of  two  planes. 

required  line  of  intersection.    A  second  auxiliary  plane  will 
determine  another  point  on  the  line  of  intersection.  The  straight 


PROBLEMS  IN  POINT,   STRAIGHT  LINE,   AND  PLANE  39 

line  joining  the  two  points  thus  found  is  the  required  line  of 
intersection  of  the  given  planes. 

The  construction  is  much  clearer  when  the  auxiliary  planes 
are  taken  parallel  to  each  other  and  perpendicular  to  one  of  the 
planes  of  projection. 

Construction.  Draw  ss  through  o  and  c  and  draw  s^s^  par- 
allel to  55  and  cutting  the  lines  of  the  given  planes.  Since  the 
plane  S^  is  perpendicular  to  H  it  cuts  the  line  PQ  from  the 
plane  of  OA  and  OB  and  the  line  MN  from  the  plane  of  CD  and 
CE.  These  lines  intersect  at  K,  a  point  of  the  required  inter- 
section. Since  the  plane  S  is  parallel  to  the  plane  Si,  the  line 
which  it  cuts  from  the  plane  of-  OA  and  OB  will  be  parallel  to 
the  line  PQ.  o'g\  parallel  to  p'q',  is  its  vertical  projection. 
Likewise,  the  line  which  the  plane  S  cuts  from  the  plane  of  CD 
and  CE  will  be  parallel  to  MN.  c'g'  is  its  vertical  projection. 
g  and  g'  are  the  projections  of  the  point  of  intersection  of  these 
two  lines.  Then  kg  and  k'g'  are  the  horizontal  and  vertical  pro- 
jections of  the  required  line  of  intersection  of  the  given  planes. 

45.  Problems. 

1.  Find  the  line  of  intersection  of  two  planes  when  one  is  oblique  to 
H  and  V  and  the  other  parallel  to  H. 

2.  Find  the  line  of  intersection  of  two  planes  when  one  is  oblique  to 
H  and  V  and  the  other  parallel  to  V. 

3.  Find  the  line  of  intersection  of  two  planes  when  one  is  oblique  to 
the  ground  line  and  one  parallel  to  the  ground  line. 

4.  Find  the  line  of  intersection  of  two  planes  when  one  is  oblique  to 
H  and  V  and  the  other  perpendicular  to  H  and  oblique  to  V. 

5.  Find  the  line  of  intersection  of  two  planes  when  each  is  given  by 
two  parallel  lines. 

6.  Find  the  line  of  intersection  of  two  planes,  each  given  by  two, lines, 
when  all  the  lines  pass  through  a  given  point  P. 

7.  Find  the  line  of  intersection  of  two  planes  when  they  are  both 
parallel  to  the  ground  line. 

8.  Find  the  line  of  intersection  of  two  oblique  planes  when  a  line  of 
one  of  the  planes  is  a  line  of  profile. 

9.  Find  the  line  of  intersection  of  an  oblique  plane  and  a  profile  plane. 
(Use  the  second  method.) 

10.  A  plane  is  given  by  two  parallel  lines  which  make  30°  with  H  and 
are  oblique  to  V  and  another  plane  is  given  by  two  parallel  lines  which 


40  DESCRIPTIVE   GEOMETRY 

make  45°  with  V  and  are  oblique  to  H.    Find  the  line  of  intersection  of 
the  planes. 

11.  A  plane  makes  45°  with  H  and  is  oblique  to  V  and  another  plane 
makes  60°  with  V  and  is  oblique  to  H.  Find  the  line  of  intersection  of 
the  planes. 

12.  Two  oblique  lines  are  given  by  their  projections.  These  are  lines 
of  greatest  slope  of  two  planes.  Find  the  line  of  intersection  of  the 
planes. 

46.  A  line  perpendicular  to  a  plane. 

A  line  which  is  perpendicular  to  a  plane  is  perpendicular  to 
every  line  of  the  plane  and  is,  therefore,  perpendicular  to  all 
horizontals  and  frontals  of  the  plane.  The  horizontal  projec- 
tion of  a  perpendicular  to  a  plane  is  at  right  angles  to  the  hor- 
izontal projection  of  any  of  its  horizontals  (Art.  36).  Like- 
wise, the  vertical  projection  of  a  perpendicular  to  a  plane  is  at 
right  angles  to  the  vertical  projection  of  any  of  its  frontals. 
Therefore,  to  draw  the  projections  of  a  perpendicular  to  a 
plane,  draw  the  projections  of  a  horizontal  and  a  frontal  of  the 
plane.  Then  the  horizontal  projection  of  the  perpendicular  to 
the  plane  is  perpendicular  to  the  horizontal  projection  of  this 
horizontal  and  its  vertical  projection  is  perpendicular  to  the 
vertical  projection  of  this  frontal. 

47.  Given  the  projections  of  two  lines  of  a  plane  and  the  pro- 
jections of  a  point  in  space,  find  the  distance  from  the  point  to 
the  plane. 

Let  AB  and  CG,  Fig.  32,  be  the  lines  which  represent  the 
plane  and  P  the  given  point. 

Analysis.  Draw  a  perpendicular  from  the  given  point  to  the 
given  plane  (Art.  46),  and  find  where  it  pierces  the  plane  (Art. 
42).  The  length  of  the  perpendicular  from  the  given  point  to 
the  piercing  point  is  the  required  distance. 

Construction.  HH  is  a  horizontal  and  FF  a  frontal  of  the 
given  plane  (Art.  17).  pe,  perpendicular  to  hhf  is  the  horizon- 
tal projection  of  the  perpendicular  and  p'e\  perpendicular  to 
f'f\  is  its  vertical  projection  (Art.  46).  The  perpendicular  PE 
pierces  the  plane  of  AB  and  CG  at  E  (Art.  42).  P^E,  the  true 
length  of  PE,  is  the  required  distance. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,   AND  PLANE 


41 


48.  Conversely,  to  find  the  projections  of  a  point  which  is  a 
given  distance  from  a  given  plane. 

Analysis.  Take  any  point  on  the  plane  and  at  this  point 
erect  a  perpendicular  of  indefinite  length  to  the  plane.  Select 
any  other  point  of  the  perpendicular  and  find  the  true  length 


FiQ.  32. — p  e  distance  from  P  to  plane  of  ABCO. 


of  the  perpendicular  from  this  point  to  the  plane.  Locate  a 
point  on  the  true  length  of  the  perpendicular  which  is  the  re- 
quired distance  from  the  point  on  the  plane.  Counter  revolve 
the  point  thus  found  to  its  position  on  the  perpendicular.  This 
is  the  required  point. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

This  problem  is  used  to  locate  the  vertex  of  a  right  cone  or 
pyramid  or  the  upper  comers  of  a  rectangular  object  when  the 
base  of  the  object  is  on  an  oblique  plane. 


42  DESCRIPTIVE   GEOMETRY 

49.  Problems. 

1.  Find  the  distance  from  a  point  to  the  plane  of  three  points. 

2.  Find  the  distance  from  a  point  to  the  plane  of  two  parallel  lines. 

3.  Find  the  distance  from  a  point  to  a  plane  which  is  parallel  to  G.  L. 

4.  A  regular  triangular  pyramid  has  its  base  parallel  to  H,  side  of 
base  3'',  altitude  i'\  Find  the  distance  from  one  corner  of  the  base  to 
the  plane  of  the  opposite  face. 

5.  Find  the  projections  of  a  point  which  is  two  units  from  a  given 
plane. 

6.  Find  a  plane  which  is  parallel  to  and  two  units  from  a  given  oblique 
plane. 

7.  Given  the  projections  of  any  four  points  A,  B,  C,  D.  These  points 
are  the  corners  of  a  tetrahedron.  Find  the  true  length  of  the  altitude 
of  the  tetrahedron  when  (a)  ABC,  (b)  ABD,  (c)  AOD,  (d)  BOD,  is  the 


8.  Find  the  distance  between  two  given  parallel  planes. 

9.  Represent  a  plane  which  contains  a  given  line  and  is  perpendicular 
to  a  given  plane.    Find  the  line  of  intersection  of  the  two  planes. 

10.  Represent  a  plane  which  makes  30'*  with  H  and  is  oblique  to  V. 
Draw  the  projections  of  a  perpendicular  to  this  plane  and  find  the  angle 
which  it  makes  with  H. 

11.  Find  the  locus  of  all  points  which  are  equidistant  from  three  given 
points. 

12.  A  plane  is  represented  by  two  of  its  lines,  and  three  points  are 
given  by  their  projections.  Find  a  point  on  the  plane  which  is  equidis- 
tant from  the  three  given  points. 

50.  Given  the  projections  of  a  point  and  of  an  oblique  line, 
represent  a  plane  which  contains  the  point  and  is  perpendicular 
to  the  line. 

Analysis.  Since  the  plane  is  to  be  perpendicular  to  the  line, 
a  horizontal  of  the  plane  will  have  its  horizontal  projection  per- 
pendicular to  the  horizontal  projection  of  the  line.  Likewise,  a 
frontal  of  the  plane  will  have  its  vertical  projection  perpendic- 
ular to  the  vertical  projection  of  the  line.  Therefore  to  rep- 
resent the  plane,  draw  the  projections  of  a  horizontal  and  of  a 
frontal  which  pass  through  the  given  point  and  are  perpendic- 
ular to  the  given  line.  These  two  lines  will  represent  the  re- 
quired plane. 


43 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

61.  Problems. 

1.  A  given  oblique  line  is  the  edge  of  a  cube.  Represent  by  two  lines 
the  plane  of  the  base  of  the  cube. 

2.  A  given  oblique  line  is  the  axis  of  a  right  pyramid.  Represent  by 
two  lines  the  plane  of  the  base  of  the  pyramid. 

3.  A  point  and  an  oblique  line  are  given  by  their  projections.  Repre- 
sent the  plane  of  the  circle  in  which  the  point  moves  when  it  is  re- 
volved about  the  line  as  an  axis. 

4.  A  given  line  AB  is  the  base  of  an  isosceles  triangle  which  has  its 
vertex  in  another  given  line  OE.    Draw  the  projections  of  the  triangle. 

5.  Represent  a  plane  which  is  twice  as  far  from  a  given  point  as  it  is 
from  a  given  plane. 

6.  Given  the  two  projections  of  one  side  of  a  square  and  the  direction 
of  the  horizontal  projection  of  an  adjacent  side.  Draw  the  projections 
of  the  square. 

7.  Given  the  two  projections  of  one  side  of  a  square  and  the  horizontal 
projection  of  a  line  which  contains  the  opposite  side.  Draw  the  pro- 
jections of  the  square. 

62.  Given  the  projections  of  two  lines  of  a  plane  and  the  pro- 
jections of  a  line  in  space,  find  the  projection  of  the  line  on  the 
given  plane. 

Analysis.  From  any  two  points  of  the  line,. erect  perpendic- 
ulars to  the  given  plane.  A  line  joining  the  points  in  which 
these  perpendiculars  pierce  the  plane  will  be  the  required  pro- 
jection. The  point  in  which  the  given  line  pierces  the  given 
plane  is  also  a  point  on  the  projection  of  the  line  on  the  plane. 

Let  the  constructon  be  made  in  accordance  with  the  above 
analysis. 

63.  Problems. 

1.  Find  the  projection  of  a  line  which  is  parallel  to  H  and  oblique  to 
V  on  a  plane  which  is  oblique  to  the  ground  line. 

2.  Find  the  projection  of  a  line  which  is  parallel  to  V  and  oblique  to 
H  on  a  plane  which  is  oblique  to  the  ground  line. 

3.  The  horizontal  and  vertical  projections  of  a  line  are  parallel  re- 
spectively to  a  horizontal  and  frontal  of  a  given  plane.  Find  the  pro- 
jection of  the  line  on  the  plane. 


44  DESCRIPTIVE   GEOMETRY 

4.  Find  the  projection  of  a  line  which  is  parallel  to  the  ground  line  on 
a  given  oblique  plane. 

5.  Find  the  projection  of  a  line  which  is  oblique  to  H  and  V  on  a  plane 
which  is  parallel  to  the  ground  line. 

6.  Find  the  projection  of  a  line  which  lies  in  a  profile  plane  on  a 
plane  which  is  oblique  to  the  ground  line. 

7.  A  line  3''  long  is  oblique  to  H  and  V.  Find  the  length  of  its  pro- 
jection on  a  plane  which  is  oblique  to  H  and  V. 

8.  A  regular  triangular  pyramid  has  its  base  parallel  to  H.  Find  the 
length  of  the  projection  of  one  of  its  lateral  edges  on  the  plane  of  the 
opposite  face'. 

9.  Project  a  given  triangle  ABC  upon  a  given  plane  MNO,  and  find 
the  true  size  of  this  new  triangle. 

10.  Given  a  plane  MNO  and  two  non-intersecting  lines  AB  and  CE 
which  are  oblique  to  the  plane.  Draw  the  projections  of  a  line  which 
is  perpendicular  to  the  plane  and  touches  AB  and  CE. 

11.  Given  a  plane  MNO  and  a  line  AB  which  is  oblique  to  the  plane. 
Draw  the  projections  of  a  line  CE  which  lies  on  the  plane  MNO  and  is 
perpendicular  to  AB. 

54.  Given  the  projections  of  two  lines  of  a  plane  and  the  pro- 
jections of  a  line  in  space,  find  the  angle  which  the  line  makes 
with  the  given  plane. 

Analysis.  The  angle  which  a  line  makes  with  a  given  plane 
is  understood  to  be  the  angle  which  the  line  makes  with  its 
projection  on  that  plane.  If  a  perpendicular  be  dropped  to  the 
plane  from  any  point  in  the  given  line,  the  angle  between  this 
perpendicular  and  the  given  line  is  the  complement  of  the  angle 
which  the  line  makes  with  the  plane.  Therefore,  find  the  angle 
between  the  given  line  and  a  perpendicular  to  the  plane  from 
any  point  of  the  line  and  construct  its  complement.  This  com- 
plement is  the  required  angle. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

55.  Problems. 

1.  Find  the  angle  which  a  line  parallel  to  the  ground  line  makes  with 
a  plane  which  is  oblique  to  the  ground  line. 

2.  Find  the  angle  which  a  line  parallel  to  H  and  oblique  to  V  makes 
with  a  plane  which  is  oblique  to  the  ground  line. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,   AND  PLANE  45 

3.  Find  the  angle  which  a  line  parallel  to  V  and  oblique  to  H  makes 
with  a  plane  which  is  oblique  to  the  ground  line. 

4.  The  projections  of  a  line  are  parallel  respectively  to  a  horizontal 
and  a  frontal  of  a  given  plane.  Find  the  angle  which  the  line  makes 
with  the  plane. 

5.  Find  the  angle  which  a  line  oblique  to  H  and  V  makes  with  a 
plane  parallel  to  the  ground  line. 

6.  Find  the  angle  which  an  oblique  line  makes  with  a  profile  plane. 

7.  Find  the  angle  which  a  line  of  profile  makes  with  a  plane  parallel 
to  the  ground  line. 

8.  Find  the  angle  which  a  line  in  a  profile  plane  makes  with  a  given 
oblique  plane. 

9.  Draw  the  projections  of  a  line  which  passes  through  a  given  point 
and  makes  30°  with  a  given  oblique  plane. 

10.  Represent  a  plane  which  contains  a  given  point  on  a  given  line 
and  makes  a  given  angle  with  the  line. , 

11.  Represent  a  plane  which  contains  a  given  point  P,  is  perpendic- 
ular to  H,  and  makes  a  given  angle  with  a  given  oblique  line. 

56.  Two  planes  are  each  given  by  the  projections  of  two  of 
their  lines,  find  the  angle  between  the  planes. 

Analysis.  From  any  point  in  space  drop  a  perpendicular  to 
each  plane  (Art.  46).  The  angle  between  these  perpendiculars 
(Art.  35),  is  the  angle  between  the  planes. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

57.  Problems. 

1.  Find  the  angle  between  two  planes  when  one  is  oblique  to  H  and  V 
and  one  perpendicular  to  H  and  oblique  to  V. 

2.  Find  the  angle  between  two  planes  when  one  is  oblique  to  H  and  V 
and  one  perpendicular  to  V  and  oblique  to  H. 

3.  Find  the  angle  between  two  planes  when  one  is  oblique  to  H  and 
V  and  one  parallel  to  H. 

4.  Find  the  angle  between  two  planes  when  one  is  oblique  to  the 
ground  line  and  one  parallel  to  the  ground  line. 

5.  Find  the  angle  between  two  planes  when  they  are  both  parallel  to 
the  ground  line. 

6.  Find  the  angle  between  a  profile  plane  and  a  plane  which  is  oblique 
to  H  and  V. 

7.  A  plane  is  represented  by  a  line  of  profile  and  a  point.  Find  the 
angle  between  this  plane  and  a  profile  plane. 


46  DESCRIPTIVE   GEOMETRY 

8.  A  regular  triangular  pyramid  has  its  base  parallel  to  H.  Find  the 
true  size  of  the  angle  between  two  lateral  faces. 

9.  In  the  pyramid  of  problem  8,  find  the  true  size  of  the  angle  between 
the  base  and  one  of  the  lateral  faces. 

10.  Given  an  oblique  line  AB  and  two  points  C  and  E,  find  the  angle 
between  the  planes  CAB  and  EAB. 

11.  Given  a  plane  parallel  to  the  ground  line  and  a  point;  represent 
a  plane  which  contains  the  point,  is  parallel  to  the  ground  line,  and 
makes  60°  with  the  given  plane. 

12.  An  oblique  plane  is  given  by  two  oblique  lines  AB  and  AC.  Rep- 
resent a  plane  which  contains  AB  and  makes  45°  with  the  plane  ABC. 

13.  A  line  AB  makes  30°  with  H  and  is  oblique  to  V.  Represent  a 
plane  ABC  which  makes  45°  with  H  and  a  plane  ABE  which  makes  60° 
with  the  plane  ABC. 

14.  Find  the  angle  between  the  faces  A  and  B  of  the  lamp  shade  shown 
in  Fig.  35,  page  49. 

15.  Find  the  angle  between  the  faces  A  and  D  of  the  ventilator  pipe 
shown  in  Fig.  36,  page  49. 

16.  Find  the  angle  between  the  bottom  C  and  the  side  A  in  the  chute 
shown  in  Fig.  38,  page  49. 

58.  Two  oblique  lines  which  do  not  lie  in  the  same  plane  are 
given  by  their  projections,  find  the  projections  and  the  true 
length  of  their  common  perpendicular. 

Let  AB  and  CK,  Fig.  33,  be  the  given  lines. 

Analysis.  Pass  a  plane  through  one  line  parallel  to  the  other 
and  project  the  second  line  on  this  plane.  This  projection  will 
be  parallel  to  the  line  itself,  since  the  line  is  parallel  to  the 
plane.  Where  this  projection  cuts  the  first  line,  erect  a  per- 
pendicular to  the  plane.  This  is  the  common  perpendicular  to 
the  two  lines. 

Construction.  Through  any  point  P  of  the  line  AB,  draw  EG 
parallel  to  CK.  Project  any  point  0  of  CK  on  the  plane  of  AB 
and  EG.  M  is  the  projection  of  this  point  and  MN,  parallel  to 
CG,  is  the  projection  of  CG  on  this  plane.  MN  intersects  AB 
at  X,  and  XY,  perpendicular  to  the  plane,  is  the  required  line. 
x^y  is  the  true  length  of  this  perpendicular. 

59.  Second  method,  when  one  of  the  lines  is  parallel  to  one  of 
the  planes  of  projection. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,   AND   PLANE 


47 


Let  AB  and  HH,  Fig.  34,  be  the  given  lines. 

Analysis.  Draw  the  projections  of  the  two  given  lines  on  an 
auxiliary  plane  which  is  perpendicular  to  one  of  the  planes  of 
projection  and  also  perpendicular  to  one  of  the  given  lines.     In 


FiQ.  33. — XY  xommon  perpendicularFm.  34. — XY  common  perpendicular 
to  AB  and  CK,  to  AB  and  HH. 

this  auxiliary  view,  one  of  the  given  lines  appears  as  a  point, 
the  other  as  a  straight  line.  A  perpendicular  from  the  point 
to  this  line  is  the  auxiliary  view  of  the  common  perpendicular 
to  the  two  given  lines  and  is  the  true  length  of  the  perpendic- 
ular. Draw  this  view  of  the  common  perpendicular  first  and 
then  by  projection  get  the  horizontal  and  vertical  views. 

Construction.  In  Fig.  34,  S  is  the  auxiliary  plane  which  is  per- 
pendicular to  H  and  also  to  the  line  HH.  When  this  plane  is 
turned  over  on  a  level  with  HH,  y^  is  the  auxiliary  view  of  HH 
and  o-i&i  the  auxiliary  view  of  AB.  x^y^,  perpendicular  to 
difei,  is  the  auxiliary  view  of  the  common  perpendicular  to  the 
two  given  lines  and  is  its  true  length,  x  is  the  horizontal  view 
of  the  point  X  and  xy,  perpendicular  to  Kk,  is  the  horizontal 


48  DESCRIPTIVE   GEOMETRY 

view  and  x'y'  is  the  vertical  view  of  the  required  common  per- 
pendicular to  the  lines  AB  and  HH. 

60.  Problems. 

1.  Find  the  common  perpendicular  to  two  lines  when  one  is  oblique 
to  H  and  V  and  the  other  is  parallel  to  H  and  oblique  to  V. 

2.  Find  the  common  perpendicular  to  two  lines  when  one  is  oblique 
to  H  and  V  and  the  other  is  parallel  to  V  and  oblique  to  H. 

3.  Find  the  common  perpendicular  to  two  lines  when  one  is  parallel 
to  H  and  oblique  to  V  and  the  other  is  parallel  to  V  and  oblique  to  H. 

4.  Find  the  common  perpendicular  to  two  lines  when  one  of  them  is 
parallel  to  the  ground  line. 

5.  Find  the  common  perpendicular  to  two  lines  when  one  of  them  lies 
In  a  profile  plane. 

6.  Find  the  common  perpendicular  to  two  oblique  lines  by  using  a 
plane  perpendicular  to  one  of  the  lines  and  projecting  the  other  line  on 
this  plane. 

7.  Given  the  projections  of  four  points  which  do  not  lie  in  one  plane, 
find  the  projections  of  a  fifth  point  which  is  equidistant  from  the  given 
points. 

8.  Represent  a  plane  which  is  parallel  to  and  equidistant  from  two 
given  lines. 

9.  Given  the  H  and  V  projections  of  a  line  AB,  the  H  projection  cd  of 
another  line  and  the  H  projection  xy  of  the  common  perpendicular  to 
the  two  lines;  find  the  V  projections  of  CD  and  XY. 

61.  Auxiliary  planes  of  projection.  The  use  of  an  auxiliary 
plane  of  projection  such  as  was  used  in  Art.  59  is  eommon  in 
commercial  practice.  The  position  of  such  a  plane  is  deter- 
mined by  the  position  of  the  object  in  space.  The  plane  is 
placed  so  that  the  projection  of  the  object,  or  part  of  the  object, 
upon  it  best  brings  out  the  particular  features  under  considera- 
tion. These  auxiliary  planes  are  usually  taken  at  right  angles 
to  one  of  the  principal  planes  of  projection. 

Figs.  35,  36,  37,  38,  also  show  the  use  of  an  auxiliary  plane  of 
projection.  Thus  in  showing  the  true  size  and  shape  of  the 
face  of  the  lamp.  Fig.  35,  the  plane  S  is  taken  perpendicular  to 
V  and  parallel  to  the  face  A  of  the  lamp.  Then,  when  the  face 
A  has  been  projected  upon  this  plane,  the  plane  is  revolved 
until  it  is  parallel  to  V  and  the  face  of  the  lamp  is  shown  in  its 
true  size  and  shape. 


PROBLEMS   IN   POINT,    STRAIGHT   LINE,    AND   PLANE  49 


Lamp  shade. 


\ 

<t^ 

0 

p 

\ 

1 

y/ 

\ 

60' 

^ 

r,% 


Hip  roof 


F^G.  35. — Lam'^  shade. 
Fig.  37.— Hip  roo/. 


Fig.  36. — Ventilator  pipe. 
Fig.  ZS.— Chute, 


50  DESCRIPTIVE   GEOMETRY 


PROBLEMS 

The  following  problems  can  be  solved  in  rectangles  10"xl4". 

The  ground  line  is  to  be  parallel  to  the  shorter  sides  of  the 
rectangle. 

The  profile  plane  P  is  in  the  center  of  the  rectangle. 

Distances  are  given  in  inches. 

Points  are  located  by  giving  their  perpendicular  distances 
from  P,  V,  and  H. 

The  first  distance  is  from  P. 

The  second  distance  is  from  V. 

The  third  distance  is  from  H. 

Distances  are  measured  to  the  right  of  P,  in  front  of  V  and 
above  H  unless  preceded  by  a  minus  sign. 

1.  Draw  the  projections  of  the  bisectors  of  the  angles  of  the  triangle 
A(— If,  — f,  -.2i),  B(— J,  — 3f,  -4),  C(2i,  —2,  ~5i). 

2.  The  horizontal  projection  of  a  parallelogram  ABCD  is  a  2^"  square. 
Draw  the  vertical  projection  of  the  parallelogram  when  its  center  is  1" 
above,  and  the  corner  B  i"  below  the  corner  A.  Find  the  true  size  of 
the  parallelogram. 

3.  Given  a  point  and  two  planes.  Draw  the  projections  of  a  line 
which  passes  through  the  point  and  is  parallel  to  both  planes. 

4.  Given  a  plane  ABC  and  a  point  P. .  Draw  a  line  through  P  so  that 
it  will  be  parallel  to  ABC  and  have  its  projections  parallel  to  each  other. 

5.  The  horizontal  projections  of  two  points  are  V  apart.  The  ver- 
tical projections  of  the  same  points  are  2"  apart.  What  is  the  greatest 
and  what  is  the  least  distance  apart  that  the  points  can  be  placed? 

6.  Two  intersecting  lines  AB  and  AC  represent  a  plane  which  makes 
60**  with  H.  ,  AB  makes  45'^  and  AC  30°  with  H.  Find  the  true  size  of 
the  angle  BAG. 

7.  Given  two  points  A  and  B,  draw  the  projections  of  the  locus  of  a 
point  which  is  Z"  from  A  and  4''  from  B. 

8.  Find  a  point  P  on  a  given  line  MN  which  is  equidistant  from  two 
given  planes  ABC  and  DEF. 

9.  Find  the  point  O  which  is  equidistant  from  the  points  A,  B,  C, 
and  D. 


PROBLEMS  51 

10.  Represent  a  plane  which  is  equidistant  from  a  point  A  and  a  line 
BC  and  is  parallel  to  another  line  DE. 

11.  Represent  a  plane  which  is  equidistant  from  any  three  points  A, 
B,  C  and  is  parallel  to  a  given  line  DE. 

12.  Through  any  three  points  A,  B,  C,  take  three  planes  which  are 
parallel  and  equidistant. 

13.  Through  any  three  points  A,  B,  C,  take  three  planes  which  are 
parallel  and  equidistant  and  such  that  one  of  the  planes  passes  through 
a  fourth  given  point  P. 

14.  Through  any  four  points  A,  B,  C,  D,  take  four  planes  which  are 
parallel  and  equidistant. 

15.  Through  any  two  points,  A  and  B,  and  through  any  given  line  CD, 
take  three  planes  which  are  parallel  and  equidistant. 

16.  Find  the  line  which  will  touch  the  three  lines:  A(— 2i,  —3,  —1) 
B(— f.  —h  — 5),  C(i,  — 1,  — 4)  D(2i,  —1,  — i)  and  E(— i,  — U,  —4 J) 
F(l. —i-l). 

17.  Through  a  given  point  take  a  line  which  touches  any  other  two 
non-intersecting  lines. 

18.  Through  three  given  lines  not  situated  two  by  two  in  planes,  take 
three  planes  which  have  a  common  line  of  intersection. 

19.  Given  a  line  AB,  a  point  P  and  a  distance  d.  Represent  a  plane 
which  contains  AB  and  is  the  distance  d  from  the  point  P. 

20.  Given  the  horizontal  projection  of  an  equilateral  triangle,  the  ver- 
tical projection  of  one  corner  and  the  true  length  of  a  side,  find  the  ver- 
tical projection  of  the  triangle. 

21.  Given  two  points  A  and  B,  a  plane  EFG  and  two  distances  d  and 
d^;  construct  a  triangle  ABC  such  that  AC=d,  BC=d  and  the  point  C 
lies  in  the  plane  EFG. 

22.  Draw  a  line  MN  which  intersects  two  given  lines  AB  and  CD  and 
is  parallel  to  and  1"  from  a  given  plane  EFG. 

23.  Draw  the  projections  of  the  triangle  A(li,  0,  — 2^)  B( — 1,  x,  — 2) 
C(y,  z,  0),  the  distances  x,  y,  and  z  being  unknown,  when  AB=2|'', 
BC=3'',  and  AC=3r'. 

24.  Draw  the  projections  of  the  center  and  diameter  of  a  sphere  which 
is  tangent  to  H  and  V  and  has  its  center  on  the  line  A( — 2,  — 2,  0) 
B(2,  0,— 3). 

25.  A  right  square  pyramid  having  an  altitude  of  2"  has  its  axis  in 
G.  L.  and  the  sides  of  the  base  parallel  to  H  or  V.  The  lateral  edges  of 
the  pyramid  make  30°  with  H  and  45"  with  V.  Show  a  profile  projec- 
tion of  the  pyramid. 

26.  At  a  point  of  outcrop  0(0,  — 2,  — 1)  the  strike  of  a  body  of  ore  is 
north  45°  east.  The  "dip"  (the  inclination  of  the  ore  body  to  the  hori- 
zontal) is  60°.    A  tunnel  is  driven  at  P(— li,  —ih  —2)  south  30°  west 


52  DESCRIPTIVE   GEOMETRY 

on  a  rising  10%  grade.    How  far  must  the  tunnel  be  driven  to  reach  the 
ore  body? 

Note.— The  "strike"  means  a  horizontal  line  in  the  plane  of  the  ore 
body. 

27.  Points  on  the  outcrop  of  an  ore  vein  have  elevations  of  A  360', 
B  480',  and  C  320'.  From  A  to  B  is  300'  due  N.,  from  B  to  C,  400'  due 
N.  B.    Find  the  strike  and  dip  of  the  vein. 

28.  Two  ore  veins  have  a  strike  due  N.  E.  and  a  dip  of  SO"*  S.  E.  The 
strike  lines  are  200'  apart.  What  is  the  shortest  distance  between  the 
veins? 

29.  From  the  point  of  outcrop  A,  run  south  30°  west  100'  to  a  drill- 
hole B.  The  drillhole  C  is  100'  from  both  A  and  B.  The  drill  strikes 
ore  at  B  at  a  depth  of  25'  and  at  C  at  a  depth  of  50'.  Determine  the  dip 
and  strike  of  the  vein. 

30.  Points  A,  B,  and  C  are  upon  a  sidehill  which  is  a  plane  dipping 
west  30°.  A  is  a  point  of  outcrop  of  the  vein.  FYom  A,  B  bears  south 
45°  west  350'  horizontally,  while  C  bears  south  15°  east  700'  on  the  slope 
of  the  hill.  At  B  and  C,  drillholes  driven  perpendicular  to  the  surface 
each  encounter  the  vein  at  175'.  Determine  the  strike  and  dip  of  the 
vein. 

31.  A  horizontal  tunnel  cuts  a  vein  at  1000'  from  the  portal.  The 
vein  strikes  north  45°  west  and  dips  75°  to  the  southwest.  The  tunnel 
is  driven  north  30°  east.  What  will  be  the  horizontal  distance  at  which 
a  tunnel  on  a  4%  grade,  driven  from  the  same  point  and  north  15° 
west,  will  cut  the  vein? 

32.  From  A,  B  bears  north  30°  east  1500';  from  A,  C  bears  south  60° 
east  1200'.  At  A,  a  vertical  borehole  cuts  a  vein  at  290';  at  B,  an  inclined 
borehole  bears  north  75°  east,  dips  75°  and  cuts  the  vein  at  515';  at  C, 
a  vertical  hole  cuts  the  vein  at  275'.  Elevation  of  A,  9800';  B,  9600'; 
C,  9900^.    Required  the  strike  and  dip  of  the  vein. 

33.  Given  three  oil  wells,  A,  B,  C:  A,  550'  above  sea  level,  1450'  deep; 
B,  325'  above  sea  level,  975'  deep;  C,  425'  above  sea  level,  1100'  deep. 
Find  the  depth  of  a  well  to  be  sunk  to  the  same  oil  vein  at  D  which  is 
613'  above  sea  level. 

34.  The  shaft  of  a  mine  runs  N.  E.  from  a  point  A  whose  elevation  is 
150'.  The  slope  of  the  shaft  from  the  vertical  is  3  to  12.  It  is  desired 
to  run  a  tunnel  to  meet  this  shaft,  starting  at  a  point  B  1000'  from  the 
shaft  entrance,  N.  15°  west  from  it,  at  an  elevation  of  100'.  What  is 
the  shortest  tunnel  that  can  be  run?  At  what  grade  and  in  what  di- 
rection from  B? 

35.  Given  the  points  A  and  B  as  in  problem  34.  A  tunnel  is  started 
from  B  with  a  5%  drop,  running  S.  30°  E.    A  vertical  shaft  is  sunk 


PROBLEMS  53 

75'  at  A.    What  will  be  tlie  length,  direction,  and  grade  of  the  shortest 
tunnel  from  the  bottom  of  the  shaft  to  meet  the  first  tunnel? 

36.  A  borehole  cuts  a  6-foot  core  from  a  vein  whose  strike  is  north 
75°  east  and  whose  dip  is  60°  to  the  northwest.  The  borehole  bears 
north  30"  east  and  dips  75°.    Required  the  true  thickness  of  the  vein. 

37.  A  borehole  bearing  north  15°  east  and  dipping  75°  intersects  a 
vein  whose  strike  is  north  45°  east.  The  true  thickness  of  the  vein  has 
been  found  to  be  3^  feet  while  the  core  from  the  borehole  shows  6  feet 
of  vein  matter.    Find  the  dip  of  the  vein. 

38.  Given  the  three  lines  A(— i,  —3,  —3)  B(— 2f,  — 4i,  —2) 
A(— i,  —3,  —3)  C(lf,  —bh  —2)  A(— i,  —3,  —3)  D(i,  0,  — li).  As- 
sume on  the  lines  in  the  order  named,  the  distances  1^'\  ^'\  and  li" 
from  A  and  through  each  point  pass  a  plane  perpendicular  to  the  line 
which  contains  the  point.  Draw  the  projections  of  the  triangular  pyra- 
mid formed  by  the  intersection  of  the  three  planes  with  each,  other  and 
with  a  horizontal  plane  through  E(0,  0,  — 4). 

39.  A  plane  is  given  by  the  three  points,  M(— 4f,  — i,  0),  N(0,  —5,  0), 
and  0(0,  —h  — 4f ).  A  square  which  has  A  (If,  —i,  0)  B(— i,  — l^s,  0) 
for  one  side  lies  in  a  horizontal  plane.  Revolve  the  square  about  MN 
as  an  axis  until  it  comes  into  the  given  plane  MNO  and  find  its  pro- 
jections in  the  new  position. 

40.  Let  the  square  in  the  preceding  problem  be  the  base  of  a  cube 
resting  on  the  given  plane.  Draw  the  horizontal  and  vertical  projec- 
tions of  the  cube. 

41.  Let  the  plane  be  given  as  in  problem  39.  A  cube  whose  edge  is 
2Y'  rests  on  this  plane  in  such  a  position  that  two  edges  AB  and  EF  of 
the  base  make  angles  of  15°  with  the  horizontal  MN.  The  corner  B  of 
the  base  Is  at  the  point  (—1,  y,  — 1).  Draw  the  horizontal  and  ver- 
tical projections  of  the  cube. 

42.  Keeping  the  edge  BA  as  in  41,  draw  the  projections  of  the  same 
cube  having  one  of  its  corners  in  H. 

43.  Keeping  the  edge  BA  as  In  41,  draw  the  projections  of  the  cube 
having  one  of  its  corners  in  a  plane  parallel  to  and  li''  from  V. 

44.  Keeping  the  edge  BA  as  in  41,  draw  the  projections  of  the  cube 
having  the  edge  BF  making  an  angle  of  30°  with  H. 

45.  Keeping  the  edge  BA  as  in  41,  draw  the  projections  of  the  cube 
having  the  edge  BF  making  an  angle  of  45°  with  V. 

46.  Keeping  the  edge  BA  as  in  41,  draw  the  projections  of  the  cube 
having  the  plane  of  its  face  ABFE  making  an  angle  of  -30°  with  the 
plane  MNO. 

47.  Keeping  the  edge  BA  as  in  41,  draw  the  projections  of  the  cube 
having  the  plane  of  its  face  ABFE  making  an  angle  of  45°  with  H. 


54  DESCRIPTIVE   GEOMETRY 

418.  Keeping  the  corner  B  as  in  41,  draw  the  projections  of  the  cube 
having  one  corner  in  H  and  another  in  V. 

49.  One  side  of  an  equilateral  triangle  lies  along  the  line  A( — 2^,  — 3, 
— i2J)  B(li,  — 1,  — 5f )  and  the  vertex  opposite  this  side  is  at  the  point 
C(i,  — 4i,  — 3).  Draw  the  projections  of  a  regular  pyramid  having 
this  triangle  for  a  base  and  an  altitude  of  4i". 

50.  Find  the  projections  of  a  regular  hexagonal  pyramid  standing  on 
the  plane  through  A( — If,  — 3,  — 2f)  parallel  to  the  lines  B( — 3^,  0, 
—^W  C(2i,  —4,  — 3i)  and  D(— li,  — 4i,  —W  E(lf,  — f,  —5).  One 
side  of  the  base  lies  in  the  line  of  intersection  of  the  plane  of  the  base 
and  a  plane  through  A  perpendicular  to  the  line  BC.  The  vertex  is  at 
the  point  0(2|,  0,  0).  Find  the  points  in  which  the  lines  BC  and  DE 
pierce  the  pyramid. 

51.  Substitute  a  regular  triangular  pyramid  for  the  regular  hexagonal 
pyramid  in  problem  50. 

52.  Substitute  a  regular  square  pyramid  for  the  regular  hexagonal 
pyramid  of  problem  50. 

53.  Substitute  a  regular  pentagonal  pyramid  for  the  regular  hexag- 
onal pyramid  of  problem  50. 

54.  Substitute  a  right  circular  cone  for  the  hexagonal  pyramid  of 
problem  50.  The  base  of  the  cone  is  tangent  to  the  line  of  intersection 
of  the  two  planes. 

55.  The  line  A(— li,  0,  —W  B(li,  —2^,  0)  coincides  with  the  edge 
of  a  cube  having  one  corner  in  H  and  another  corner  in  V.  Edge  of 
cube  3''.    Draw  the  projections  of  the  cube. 

56.  Draw  the  projections  of  the  cube  of  problem  55,  when  one  edge 
coincides  with  AB,  one  corner  is  at  A  and  the  comer  at  the  other  end 
of  the  diagonal  through  A  also  lies  in  V. 

57.  Draw  the  projections  of  the  cube  of  problem  55,  when  one  edge 
coincides  with  AB,  one  corner  is  at  A,  and  the  corner  at  the  other  end 
of  the  diagonal  of  a  face  from  A  also  lies  in  V. 

58.  The  same  as  problem  55  when  one  corner  lies  at  B  and  some  other 
corner  is  also  in  H  but  no  corner  is  in  V. 

59.  The  same  as  problem  55  when  neither  A  nor  B  is  a  corner  of  the 

cube. 

60.  Ci—lh  —If,  —W  Edi,  —3^,  — 4i)  is  a  diagonal  of  a  cube. 
One  corner  of  the  cube  is  in  a  plane  which  makes  60°  with  H  and  con- 
tains a  horizontal  passing  through  the  point  P(— 3f,  0,  0)  and  making 
45°  with  V.    Draw  the  projections  of  the  cube. 

61.  C(— 3i,  —Ih  —U)  E(1J,  —n,  — li)  is  a  diagonal  of  a  cube  which 
has  one  corner  in  H.    Draw  the  projections  of  the  cube. 

62.  The  same  as  problem  61,  but  having  one  corner  of  the  cube  in  V 
In  the  place  of  in  H. 


PROBLEMS  55 

63.  One  edge  of  a  2V'  cube  lies  along  the  line  A(— 2i,  —31,  —4) 
B(li,  0,  — 2f)  and  an  adjacent  edge  passes  through  the  point 
P(i,  — li,  x)  in  such  a  direction  that  its  horizontal  projection  makes 
60°  with  the  horizontal  projection  of  AB.  Draw  the  projections  of  the 
cube. 

64.  The  face  of  the  cube  in  problem  63  which  lies  on  the  plane  ABP 
is  the  base  of  a  right  square  pyramid.  Draw  the  projections  of  the 
pyramid  when  it  has  an  altitude  of  3i". 

65.  The  shortest  line  between  the  lines  A(— i,  — 3^,  0)  B(|,  0,  --21) 
and  C(— 2i,  —2,  —1)  D(J,  — 2f,  —3)  is  the  axis  of  a  right  rectangular 
prism.  The  diagonals  of  the  upper  and  lower  bases  coincide  with  AB 
and  CD  respectively.  The  shorter  edge  of  the  base  is  If'  long.  Draw 
the  projections  of  the  prism. 

66.  Substitute  A(3^,  0,  —If)  B(3i,  —2,  — 3f)  and  C(— 2,  —2h  —W 
D(li,  — 5,  0)  in  problem  65. 

67.  Given  a  line  AB,  two  points  C  and  E  and  the  distance  d.  Find  a 
line  which  passes  through  C,  intersects  AB  and  passes  at  the  distance  d 
from  E. 

68.  Find  OK,  the  shortest  line  which  is  parallel  to  H  and  terminates  in 
the  lines  CD  and  EF. 

69.  Project  a  cube  of  1^'  edge  on  a  plane  perpendicular  to  one  of  its 
diagonals.  (Assume  the  first  position  of  the  cube  so  that  one  of  its 
diagonals  shall  be  parallel  to  one  of  the  planes  of  projection.) 

70.  A  V  square  stick  has  its  edges  maiking  30°  to  V  and  oblique  to  H. 
Find  the  true  size  of  the  section  of  this  stick  by  the  V  and  P  planes. 

71.  Given  the  base  ABC  of  a  triangular  pyramid,  also  the  length  of 
two  edges  DA  and  DB  and  the  altitude  DO.  Draw  the  projections  of 
the  pyramid. 

72.  Draw  the  projections  of  a  tetrahedron  two  edges  of  which  are 
3",  two  4f',  and  two  5".    (No  edge  is  to  be  parallel  to  either  H  or  V. 

73.  Find  the  projections  of  three  spheres,  each  tangent  to  the  other 
two  and  having  the  following  radii:  iy\  2'\  2\'\  Construct  the  pro- 
jections of  a  fourth  sphere  with  a  radius  of  \"  which  will  be  tangent  to 
the  other  three  spheres.  The  plane  of  the  centers  of  any  three  spheres 
is  not  to  be  parallel  or  perpendicular  to  either  H  or  V. 

74.  Draw  the  projections  of  a  tetrahedron  the  sides  of  whose  base  are 
3i",  Z\'\  and  ^"  and  the  planes  of  whose  faces  make  angles  of  45°,  60°, 
and  75°  with  the  plane  of  the  base.  No  two  of  the  corners  are  to  be  the 
same  distance  from  H. 

75.  Given  an  acute  triangle  on  a  horizontal  plane.  Find  a  point  in 
space  which  joined  with  the  vertices  of  the  triangle  will  form  a  tri- 
rectangular  tetrahedron. 


56  DESCRIPTIVE   GEOMETRY 

76.  Draw  the  prajections  of  a  3"  cube  having  given  the  direction  of 
the  horizontal  projection  of  three  concurrent  edges  of  the  cube. 

77.  Draw  the  projections  of  an  equilateral  triangle  having  given 
both  projections  of  one  side  and  the  direction  of  the  horizontal  projec- 
tion of  one  of  the  other  sides. 

78.  Having  given  the  two  projections  ab  and  a'h'  of  one  edge  of  a 
cube  and  the  direction  dc  of  the  horizontal  projection  of  another  edge. 
Draw  the  projections  of  the  cube. 

79.  Having  given  the  directions  of  the  projections  of  three  concurrent 
edges  of  a  parallelopiped,  also  the  direction  and  length  of  the  diagonal 
through  the  same  corner,  construct  the  projections  of  the  parallelopiped. 

80.  Draw  the  projections  of  a  rectangular  parallelopiped,  having 
given  one  corner,  aa',  the  horizontal  projection  a&  of  one  edge,  and  the 
directions  ac  and  ad  of  the  horizontal  projections  of  two  other  edges. 

81.  A  ray  of  light  from  a  point  P,  which  is  3''  from  H  and  2"  from  V, 
is  reflected  from  a  point  in  V  which  is  11"  from  H  and  2"  to  the  right 
of  P.    Where  will  the  ray  meet  H? 

82.  A  ray  of  light  from  the  point  A  is  reflected  by  the  H  plane  at  B 
to  the  point  C  in  V,  locate  C. 

83.  Given  two  points  O  and  P  in  the  first  quadrant.  Find  the  pro- 
jections of  a  ray  of  light  which  emanates  from  O  and  after  being  re- 
flected from  both  H  and  V  passes  through  P. 

84.  Find  the  projections  of  a  ray  of  light  which  emanates  from  a 
given  point  and  is  reflected  from  H,  V,  and  P. 

85.  A  ray  of  light  from  A  is  reflected  at  B  to  C.  Find  the  reflecting 
plane  T. 

86.  Represent  a  plane  which  passes  through  a  given  point  and  makes 
given  angles  with  the  planes  of  projection.  Between  what  limits  may 
the  sum  of  these  angles  vary? 

87.  Given  a  line  AB  which  is  oblique  to  H  and  V.  Represent  a  plane 
which  contains  the  line  AB  and  is  in  such  a  position  that  AB  bisects 
the  angle  between  a  horizontal  and  a  frontal  of  the  plane.  Let  the 
horizontal  and  frontal  pass  through  the  point  A. 

88.  Given  two  lines  AB  and  CD  in  space  which  do  not  intersect. 
Draw  a  third  line  which  makes  30°  with  AB  and  45°  with  OD  and  in- 
tersects them  both. 

89.  Given  the  projections  of  a  line  which  makes  30°  with  H  and  is 
oblique  to  V.  Draw  the  projections  of  a  horizontal  which  makes  45° 
with  this  line. 

90.  Draw  the  projections  of  a  regular  tetrahedron  having  given  the 
projections  a&  and  a'6'  of  one  edge  and  the  direction  ac  of  the  horizon- 
tal projection  of  another  edge. 


CHAPTER  III 

APPLICATIONS  OF  THE  ELEMENTARY  PRINCIPLES  OF 
THE  POINT,  STRAIGHT  LINE,  AND  PLANE 


SHADES  AND  SHADOWS 

62.  In  order  that  the  projections  of  an  object  shall  more 
nearly  represent  the  object  as  it  appears,  the  shadows  of  dif- 
ferent parts  of  the  object  on  other  parts  or  on  the  planes  of 
projection  are  sometimes  shown.  Shadows  are  used  more  in 
architecture  than  any  other  branch  of  engineering  but  every 
engineer  should  know  how  to  find  the  shadow  of  a  simple  ob- 
ject. Only  the  elementary  principles  of  the  subject  will  be  out- 
lined here. 

63.  Definitions.  If  an  opaque  object  is  placed  between  a 
surface  and  the  source  of  light,  the  object  will  cast  a  shadow 
upon  the  surface.  This  is  due  to  the  fact  that  the  object  inter- 
cepts the  rays  of  light  and  prevents  their  reaching  the  surface 
upon  which  the  shadow  is  cast.  When  an  opaque  body  is  sub- 
jected to  rays  of  light,  that  portion  of  the  surface  of  the  body 
which  is  turned  away  from  the  source  of  light  and  which,  there- 
fore, does  not  receive  any  of  the  rays  of  light,  is  said  to  be  in 
shade.  The  line  which  separates  the  portion  of  the  surface  of 
an  object  which  is  turned  towards  the  light  from  that  portion 
which  is  turned  away  from  the  light  is  called  the  shade  line. 
The  shadow  of  the  shade  line  of  an  object  on  a  given  surface  is 
the  outline  of  the  shadow  of  the  object  on  that  surface. 

The  source  of  light  is  the  sun  and  because  of  its  distance 
from  the  earth  the  rays  of  light  are  assumed  to  be  parallel. 
For  the  sake  of  uniformity,  the  direction  of  a  ray  of  light  is  the 
direction  of  the  diagonal  of  a  cube  which  joins  the  upper,  left 


58  DESCRIPTIVE   GEOMETRY 

hand,  front  corner  with  the  lower,  right  hand,  back  corner  when 
the  cube  has  two  faces  parallel  to  V  and  two  parallel  to  H.  In 
other  words,  a  ray  of  light  is  considered  as  coming  over  the  left 
shoulder  so  that  its  projections  make  angles  of  45°  with  the 
ground  line. 

64.  Shadows  of  points  and  lines.  To  find  the  shadow  of  a 
point  on  a  surface,  pass  a  ray  of  light  through  the  point  and 
find  where  this  ray  pierces  the  surface.  This  piercing  point  is 
the  required  shadow  point.  To  find  the  shadow  of  a  straight 
line  on  a  plane,  find  the  shadows  of  two  points  of  the  line  on 
the  plane  and  join  th6m  by  a  straight  line.  If  the  shadow  of 
the  straight  line  falls  upon  two  or  more  planes,  find  the  shadows 
of  two  points  of  the  line  on  each  plane  and  join  the  correspond- 
ing shadow  points.  The  shadow  of  the  straight  line  is  a  broken 
line  which  changes  direction  on  the  line  of  intersection  of  the 
planes  upon  which  the  shadow  is  cast.  To  find  the  shadow  of  a 
straight  line  on  a  curved  surface,  find  the  shadows  of  several 
points  of  the  line  and  join  these  shadow  points  in  the  proper 
order.  To  find  the  shadow  of  a  curved  line  on  a  surface,  it  is 
usually  necessary  to  find  the  shadows  of  several  points  of  the 
line  on  the  surface  and  then  join  the  shadow  points  in  the 
proper  order. 

The  shadow  of  a  plane  figure  on  a  plane  which  is  parallel  to 
the  plane  of  the  figure  is  the  same  size  and  form  as  the  figure 
itself. 

65.  To  find  the  shadow  of  a  square  prism  on  the  plane  of  the 
base. 

Let  the  prism  be  given  as  shown  in  Fig.  39. 
Analysis.  Since  the  shadow  of  the  prism  is  to  be  cast  upon 
the  plane  of  the  base,  it  is  evident  that  the  base  of  the  prism 
coincides  with  its  own  shadow.  The  shadow  of  the  top  is  found 
by  passing  rays  of  light  through  the  points  A,  B,  C,  and  D  and 
finding  where  these  rays  pierce  the  plane  of  the  base.  The 
straight  lines  joining  these  piercing  points,  in  the  proper  order, 
will  form  the  outline  of  the  shadow  of  the  top.  The  shadows 
of  the  comers  of  the  upper  base  joined  with  the  shadows  of 


SHADES  AND  SHADOWS 


59 


the  corresponding  corners  of  the  lower  base  will  be  the  shadows 
of  the  lateral  edges  (Art.  64).  This  completes  the  outline  of 
the  shadow  of  the  prism. 


aa 


1         1  "V 

'dd, 

1 
1 

1 

I 

1 
1 

^7         JTdl        ^l^s       y^s        Ci 
Fig.  39. — Shadow  of  square  prism. 

Construction,  hhs  is  the  horizontal  and  h'h's  is  the  vertical 
projection  of  a  ray  of  light  which  passes  through  B.  This  ray 
pierces  the  plane  of  the  base  at  Bg.  Then  Bg  is  the  shadow  of 
the  point  B  upon  the  plane  of  the  base  of  the  prism.  In  like 
manner  Cs,  Ds,  and  As  are  the  shadows  cast  by  C,  D,  and  A 
upon  the  plane  of  the  base  of  the  prism.  Joining  these  points 
in  order  gives  the  shadow  of  the  top  on  the  plane  of  the  base. 


60  DESCRIPTIVE   GEOMETRY 

Joining  Bg  with  B^  gives  the  shadovr  of  the  vertical  edge  BB^. 
The  shadows  of  the  other  vertical  edges  are  found  in  the  same 
manner.  Then  IJj^c^d^d^  is  the  horizontal  projection  of  the 
outline  of  the  shadow  of  the  prism  on  the  plane  of  the  base  of 
the  prism. 

The  face  CCiDiD  of  the  object  is  turned  away  from  the  light 
and  is  therefore  in  the  shade.  This  face  is  visible  in  the  ver- 
tical view  and  the  fact  that  it  is  in  the  shade  is  indicated  as 
shown  in  the  figure. 

From  the  figure,  it  is  seen  that  a  line  and  its  shadow  are  par- 
allel when  the  line  is  parallel  to  the  plane  upon  which  the 
shadow  is  cast. 

66.  To  find  the  shadow  of  an  object  on  the  plane  of  the  base 
and  on  the  object  itself. 

Let  the  object  be  given  as  shown  in  Fig.  40. 

This  problem  illustrates  the  finding  of  the  shadow  which  an 
opaque  object  will  cast  upon  itself  as  well  as  the  plane  of  the 
base.  It  also  shows  the  method  of  finding  the  shadow  on  a 
vertical  plane  in  addition  to  finding  the  shadow  on  a  horizontal 
plane. 

Analysis.  All  of  the  edges  shown  are  straight  lines.  If  all 
of  the  shadow  of  any  edge  falls  on  one  plane,  it  will  be  neces- 
sary to  find  the  shadow  of  only  two  points  of  the  edge  and  join 
them  by  a  straight  line.  If  the  shadow  of  an  edge  falls  on 
more  than  one  plane,  two  shadow  points  should  be  found  on 
each  plane  and  joined  by  a  straight  line.  The  shadow  of  any 
point  of  an  edge  falls  upon  the  plane  which  is  first  pierced  by 
the  ray  of  light  passing  through  that  point  (Art.  64) . 

Construction.     The    horizontal    projection    of   the    shadow 

OiB^ises  '^  which  the  object  casts  upon  the  plane  of  the 

base  is  found  as  in  Art.  65.  For  this  shadow  it  is  only  neces- 
sary to  find  where  the  rays  of  light  pierce  a  horizontal  plane. 
The  shadow  which  the  upper  part  of  the  object  casts  upon  the 
overhang  EGK  is  found  in  the  same  manner  as  the  shadow  on 
the  plane  of  the  base,  EGK  being  a  horizontal  plane. 


SHADES  AND   SHADOWS 


61 


Due  to  the  overhang  EiMiKi,  there  is  a  shadow  cast  upon  the 
vertical  faces  of  the  object.  To  find  this  shadow,  pass  a  ray  of 
light  through  the  point  0  and  find  where  it  pierces  the  vertical 
face  AD  of  the  object.     Since  the  face  is  vertical,  the  horizontal 


a/    d;o/sii. 


Fig.  40. — Shadow  of  object  on  plane  of  hase 
and  object  itself. 


projection  of  this  point  is  Os  and  the  vertical  projection  is  the 
shadow  o'e.  In  like  manner  the  shadows  of  the  points  P,  Mi, 
and  N  are  found.  Joining  these  shadow  points  with  straight 
lines  forms  the  outline  of  the  shadow  of  the  overhang  EiMiK^ 
on  the  vertical  faces  of  the  object. 


62 


DESCRIPTIVE   GEOMETRY 


67.  To  find  the  shadow  of  a  cylindrical  column  and  cap  on  a 
horizontal  plane  and  also  the  shadow  of  the  cap  on  the  column. 

Let  the  column  and  cap  be  given  as  shown  in  Fig.  41. 
Analysis.    Since  the  upper  surface  of  the  cap  is  a  horizontal 


Fig.  41. — ^Shadow  oj  column  and  cap. 


circle,  its  shadow  on  a  horizontal  plane  is  a  circle  of  the  same 
radius  as  the  cap.  The  shadow  of  the  center  of  the  circle  is 
the  center  of  the  shadow  of  the  circle.  Finding  the  shadow  of 
the  center  and  drawing  a  circle  with  a  radius  equal  to  the 
radius  of  the  cap,  will  give  the  shadow  of  the  top  of  the  cap  on 
a,  horizontal  plane.  The  shadow  of  the  lower  surface  of  the  cap 
on  a  horizontal  plane  is  found  in  the  same  way.     Common  tan- 


SHADES   AND  SHADOWS  63 

gents  to  these  circles  drawn  in  the  direction  of  the  horizontal 
projection  of  a  ray  of  light  completes  the  shadow  of  the  cap  on 
the  horizontal  plane.  The  shadow  of  the  column  on  a  horizon- 
tal plane  is  bounded  by  two  lines  drawn  in  the  direction  of  the 
horizontal  projection  of  a  ray  of  light  and  tangent  to  the  base 
of  the  column. 

To  find  the  shadow  of  the  cap  on  the  column,  pass  rays  of 
light  through  points  of  the  lower  edge  of  the  cap  and  find 
where  they  pierce  the  column.  A  line  joining  these  points  is 
the  boundary  of  the  shadow  of  the  cap  on  the  column. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

The  side  of  the  column  and  cap  turned  away  from  the  source 
of  light  is  in  the  shade.  GK  is  the  shaded  portion  of  the  col- 
umn which  is  visible  in  the  vertical  view  and  EM  is  the  shaded 
portion  of  the  cap  which  is  visible  in  the  vertical  view. 

68.  Problems. 

1.  One  edge  of  a  1"  cube  lies  on  a  horizontal  plane  and  is  oblique  to 
V.  The  plane  of  one  face  of  the  cube  makes  30°  with  H.  Draw  the  pro- 
jections of  the  cube  and  find  its  shadow  on  a  horizontal  plane. 

2.  Draw  the  projections  of  a  2"  cube  when  one  diagonal  is  perpendic- 
ular to  H.  Find  the  shadow  of  the  cube  on  a  horizontal  plane  which 
passes  through  the  lower  end  of  the  diagonal. 

3.  A  right  square  P3^aniid  has  its  vertex  in  a  horizontal  plane.  The 
side  of  the  base  is  1"  and  the  altitude  is  Z"  long,  makes  60°  with  H  and 
is  oblique  to  V.  Draw  the  projections  of  the  pyramid  and  find  Its 
shadow  on  a  horizontal  plane. 

4.  Draw  the  projections  of  a  sphere  which  has  a  radius  of  \\'\  Find 
at  least  twelve  points  In  Its  shadow  on  a  horizontal  plane. 


64 


DESCRIPTIVE    GEOMETRY 


PLANE  SECTIONS  AND  DEVELOPMENTS  OF  THE  SURFACES  OF 
PRISMS  AND  PYRAMIDS 

69.  To  find  the  line  of  intersection  of  a  right  prism  with  a 
given  oblique  plane  and  to  develop  the  surface  of  the  prism. 

Let  the  prism  be  given  as  in  Fig.  42  and  let  MN  and  OP  rep- 
resent the  oblique  plane. 

Analysis.  Find  the  points  in  which  the  lateral  edges  of  the 
prism  intersect  the  plane  of  MN  and  OP.  Straight  lines  join- 
ing these  points  of  intersection  is  the  line  of  intersection  of  the 
prism  with  the  oblique  plane. 


h,  c;  aj  fj  -J 

Pig.  42. — Plane  section  and  development  of  right  prism. 

Construction.  AAj  and  CCi  have  a  common  horizontal  pro- 
jecting plane  which  cuts  the  line  MN  at  E  and  the  line  OP  at  K. 
e'liff  the  vertical  projection  of  the  line  joining  these  points,  cuts 
the  vertical  projections  of  the  edges  of  the  prism  at  x'  and  z^ 
which  are  the  vertical  projections  of  two  of  the  required  points. 
The  line  OP  cuts  another  face  of  the  prism  at  R.  x'r'  extended 
cuts  h'h'^  at  t/  which  is  the  vertical  projection  of  the  point  in 


PLANE  SECTIONS  AND  DEVELOPMENTS  OP   SURFACES  65 

which  BBi  cuts  the  plane  of  MN  and  OP.  Then  x'y'z'  is  the 
vertical  and  xyz  the  horizontal  projection  of  the  required  line 
of  intersection. 

To  develop  the  surface  of  the  prism  showing  its  line  of  inter- 
section with  the  oblique  plane.  If  one  of  the  lateral  faces  of 
the  prism  be  placed  against  a  plane  and  then  the  prism  be  re- 
volved about  an  edge  of  this  face  until  the  next  face  comes  into 
this  plane,  and  the  process  be  repeated  until  each  face  in  turn  is 
brought  into  the  plane,  the  part  of  the  plane  covered  by  the 
different  faces  of  the  prism  will  be  the  development  of  the 
prism.  In  Fig.  42,  (J3&3=ai&i,  &3C3=&iCi,  etc.  To  get  the  line 
of  intersection  on  the  development,  lay  off  a^x^=a\x\  hzy2==b\y'y 
etc.,  and  join  these  points  by  straight  lines. 

Note.  All  the  problems  in  plane  sections  of  prisms  and  pyra- 
mids can  be  multiplied  by  changing  the  number  of  lateral  faces 
of  the  surface. 

70.  To  find  the  line  of  intersection  of  an  oblique  prism  by  a 
plane  perpendicular  to  the  lateral  edges,  and  to  develop  the 
lateral  surface  of  the  prism. 

Let  the  prism  and  plane  be  given  as  in  Fig.  43. 

By  finding  the  points  in  which  the  lateral  edges  of  the  prism 
pierce  the  plane  of  HH  and  FF  and  joining  these  points  in  the 
proper  order,  the  intersection  XYZ  is  found. 

To  develop  the  lateral  surface  of  the  prism.  Since  the  sec- 
tion plane  is  perpendicular  to  the  edges  of  the  prism,  it  is  evi- 
dent that  the  line  of  intersection  will  be  a  straight  line  in  the 
development.  The  true  size  of  this  section  is  x"y''z'\  This  is 
found  by  revolving  it  until  it  is  parallel  to  the  horizontal  plane 
about  HH  as  an  axis.  In  the  development,  lay  off  x^y^=^x"y[\ 
yzZ^=y"z'\  etc.,  along  the  straight  line  x^x^.  On  a  line  perpen- 
dicular to  x^Xz  at  the  point  x^,  lay  off  below  the  line  the  true 
length  x^a-i_  of  the  edge  from  X  to  the  lower  base  of  the  prism 
and  along  the  same  line  lay  off  above  x^x^  the  true  length  x^a^ 
of  the  edge  from  X  to  the  upper  base  of  the  prism.    In  a  similar 


66 


DESCRIPTIVE   GEOMETRY 


manner,  lay  off  at  the  points  2/3,  Z3  and  x^  the  true  lengths  of  the 
edges  through  the  points  Y,  Z,  and  X.  In  this  way  all  points 
in  both  bases  of  the  prism  are  located  in  the  development. 


/     .7 


N*^ 


^ 


^ 


Fig.  43. — Plane  section  and  development  of  oblique  prisrn^ 


Straight  lines  joining  in  order  the  points  in  the  development  of 
the  lower  base  and  also  in  the  same  way  those  of  the  upper  base 
complete  the  development  of  the  prism. 

71.  To  find  the  line  of  intersection  of  an  oblique  pyramid  by  a 
plane  and  to  develop  the  lateral  surface  of  the  pyramid. 

Let  the  plane  and  pyramid  be  given  as  in  Fig.  44. 

By  finding  the  points  in  which  the  lateral  edges  of  the  pyra- 
mid pierce  the  plane  of  MN  and  OP  and  joining  these  points  in 
the  proper  order,  the  intersection  XYZ  is  found. 

To  develop  the  lateral  surface  of  the  pyramid.  If  the  face 
VAB  of  the  pyramid  be  placed  against  a  plane  and  the  pyramid 
be  rolled  over  about  VB  as  an  axis  until  the  face  YBC  comes 


PLANE  SECTIONS  AND  DEVELOPMENTS  OF   SUEPACES 


67 


into  the  plane  and  then  the  pyramid  rolled  again  about  VC  as 
an  axis  until  the  next  face  comes  into  the  plane,  the  part  of  the 
plane  thus  covered  is  the  development  of  the  pyramid.  The 
development  is  found  by  constructing  a  series  of  triangles  hav- 
ing given  the  length  of  the  sides  of  each  triangle.  The  first 
triangle  is  formed  of  the  sides  VA,  VB,  and  AB ;  the  second  of 
the  sides  YB,  VC,  and  BC,  etc. 


a'  y       0' 

Fis.  44. — Flane  section  and  development  of  otlique  pyramid. 

The  line  of  intersection  of  the  pyramid  with  the  plane  of  MN 
and  OP  can  be  layed  off  on  the  development  by  laying  off  on 
each  edge  in  succession  the  true  lengths  of  VX,  VY,  and  VZ 
and  joining  these  points  by  straight  lines. 

72.  To  find  the  line  of  intersection  of  any  surface  with  a 
given  oblique  plane  by  using  an  auxiliary  plane  of  projection. 

Fig.  45  represents  an  oblique  prism  cut  by  the  plane  of  HH 
and  FF.  The  auxiliary  plane  of  projection  is  taken  at  right 
angles  to  the  horizontal  HH  of  the  given  plane.  In  the  auxil- 
iary view,  the  given  plane  is  represented  by  the  line  x^Zz  and 


68 


DESCRIPTIVE    GEOMETRY 


the  points  in  which  the  lateral  edges  of  the  prism  pierce  this 
plane  by  x^,  2/2,  ^z-  By  projection,  the  horizontal  view  xyz  and 
the  vertical  view  x'yfz'  of  the  intersection  are  found.     If  any 


"VPz 


a,'        cT 

Fig.  45. — Plane  section  of  oblique  prism  by 
using  auxiliary  plane  of  projection. 


element  of  the  surface  is  parallel  to  the  auxiliary  plane  of  pro- 
jection, its  view  on  that  plane  is  the  true  length  of  the  element. 
The  auxiliary  plane  of  projection  should  always  be  taken  at 
right  angles  to  a  horizontal  or  a  frontal  of  the  plane  of  the 
section. 


PLANE  SECTIONS  AND   DEVELOPMENTS  OP   SURFACES  69 

73.  Problems. 

1.  Assume  any  four  points,  which  are  not  in  the  same  plane,  and 
draw  through  them  four  parallel  lines  in  such  a  way  that  the  plane 
section  of  the  prism,  which  has  the  four  lines  for  edges,  will  be  a  par- 
allelogram. 

2.  Given  a  point  on  the  edge  of  any  tetrahedron.  Find  the  shortest 
path  to  be  followed  on  the  faces  in  order  to  go  around  the  tetrahedron 
and  return  to  the  same  point.  The  path  is  to  cross  the  three  faces  but 
not  the  base. 

3.  Two  intersecting  planes  cut  the  ground  line  at  the  points  T  and  S. 
Find  the  projections  of  the  shortest  path  on  the  planes  from  T  to  S. 

4.  Pass  a  plane  so  as  to  cut  a  regular  hexagon  from  a  3"  cube. 

5.  Cut  a  parallelogram  having  one  side  2"  long  from  the  pyramid 
O— ABCD.  0(U,  — 5i,  0),  A(— 3,  —1,  — 5i),  B(i,  0,  — 5i),  C(2i, 
—2,  — 5i),  D(— If,  — 4i  —51). 

6.  Given  a  square  prism  with  its  axis  perpendicular  to  H.  Side  of 
base  2i".  Cut  a  parallelogram  from  the  prism  having  one  side  3"  long 
and  one  angle  60°. 

7.  Given  a  square  prism  with  its  axis  perpendicular  to  H.  Side  of 
base  2i".  Cut  a  parallelogram  from  the  prism  having  one  diagonal  6i" 
long  and  one  angle  60°. 


70 


DESCRIPTIVE    GEOMETRY 


INTERSECTIONS  OF  THE  SURFACES  OF  PRISMS  AND  PYRAMIDS 

74.  To  find  the  line  of  intersection  of  the  surfaces  of  two 
prisms  when  the  lateral  edges  of  one  of  them  is  perpendicular 
to  the  horizontal  plane. 

Let  the  prisms  be  given  as  in  Fig.  46. 

Analysis,  Since  the  prisms  have  plane  faces,  it  is  only  nec- 
essary to  find  the  points  in  which  the  edges  of  each  prism  pierce 


p'  i^/     q'    xizl 


Fig.  46. — Intersection  of  prisms. 


the  faces  of  the  other  prism.  A  series  of  straight  lines  joining 
in  order  these  piercing  points  is  the  required  line  of  intersec- 
tion. 

Consfruction.  It  is  evident  from  the  drawing  that  A,  B,  C, 
D,  E,  and  M  are  the  points  in  which  the  edges  of  the  triangular 
prism  pierce  the  faces  of  the  square  prism.  It  is  also  evident 
that  three  of  the  edges  of  the  square  prism  do  not  cut  the  faces 
of  the  triangular  prism.    To  find  the  points  in  which  the  fourth 


INTERSECTIONS   OF   SURFACES 


71 


edge  of  the  square  prism  intersects  the  triangular  prism,  pass  a 
plane  T  through  this  edge  and  parallel  to  the  lateral  edges  of 
the  triangular  prism.  This  plane  cuts  two  lines  from  the  tri- 
angular prism  which  intersect  the  edge  of  the  square  prism  in 
the  points  K  and  G.  These  are  the  points  in  which  the  edge  of 
the  square  prism  intersects  the  faces  of  the  triangular  prism. 
Joining  all  of  the  piercing  points  in  the  proper  order  gives  the 
intersection  as  shown  in  Fig.  46. 

75.  To  find  the  line  of  intersection  af  the  surfaces  of  two  ob- 
lique prisms. 

Let  the  prisms  be  given  as  in  Fig.  47. 


w      X'    z'      y'  n'  q^ 

Fig.  47. — Intersection  of  bdlique  prisms. 


72  DESCRIPTIVE   GEOMETRY 

Analysis.  Pass  a  system  of  auxiliary  planes  parallel  to  the 
lateral  edges  of  both  prisms.  These  planes  will  cut  straight 
lines  from  each  prism  which  will  intersect  in  points  of  the  re- 
quired line  of  intersection. 

Construction.  Through  any  point  P,  draw  lines  parallel  to 
the  lateral  edges  of  each  prism,  tt  is  the  horizontal  projection 
of  the  line  of  intersection  of  the  plane  of  these  lines  with  the 
plane  of  the  bases  of  the  prisms.  Each  of  the  auxiliary  planes 
cuts  a  line  from  the  plane  of  the  bases  of  the  prisms  which  is  par- 
allel to  tt.  The  planes  also  cut  lines  from  the  prisms  which  are 
parallel  to  the  respective  edges  of  the  prisms  and  which  in- 
tersect in  points  of  the  required  line  of  intersection.  The  line 
of  intersection  is  a  broken  line  which  only  changes  direction  at 
the  points  where  the  edges  of  one  prism  pierce  the  faces  of  the 
other.  By  using  only  the  planes  which  contain  edges  of  one  or 
the  other  of  the  prisms  and  joining  in  order  the  points  thus 
found,  the  broken  line  of  intersection  as  shown  in  Fig.  47  is 
found. 

76.  To  find  the  line  of  intersection  of  the  surfaces  of  a  prism 
and  a  pyramid. 

Let  the  prism  and  the  pyramid  be  given  as  in  Fig.  48. 

Analysis.  Pass  a  system  of  auxiliary  planes  through  the  ver- 
tex of  the  pyramid  and  parallel  to  the  lateral  edges  of  the  prism. 
These  planes  will  cut  straight  lines  from  each  surface  which 
will  intersect  in  points  of  the  required  line  of  intersection.  A 
line  through  the  vertex  of  the  pyramid  and  parallel  to  the  lat- 
eral edges  of  the  prism  will  lie  in  all  of  the  auxiliary  planes  and 
will  therefore  pierce  the  plane  of  the  bases  in  a  point  common 
to  all  the  lines  cut  from  the  plane  of  the  bases. 

Construction.  Through  V^  draw  YiS  parallel  to  the  lateral 
edges  of  the  prism.  It  pierces  the  plane  of  the  bases  at  S. 
Through  5,  draw  st,  st^,  stz,  lines  which  the  auxiliary  planes  cut 
from  the  plane  of  the  bases.  The  intersection  of  the  lines  which 
these  planes  cut  from  the  prism  and  pyramid  are  points  on  the 
required  line  of  intersection.     By  using  only  the  planes  which 


INTERSECTIONS   OF   SURFACES 


73 


contain  edges  of  the  prism  or  the  pyramid  and  joining  in  order, 
with  straight  lines,  the  points  thus  found  the  line  of  intersection 
as  shown  in  Fig.  48  is  found. 

Ha 


F^G.  48. — Intersection  of  prism  and  pyramid. 

77.  To  find  the  line  of  intersection  of  the  surfaces  of  two 
pjrramids. 

Let  the  pyramids  be  given  as  in  Fig.  49. 

Analysis.  Pass  a  system  of  auxiliary  planes  through  the  ver- 
tices of  the  pyramids.  These  planes  will  cut  straight  lines  from 
each  pyramid  which  will  intersect  in  points  of  the  required  line  of 
intersection.     A  line  joining  the  vertices  of  the  pyramids  will  lie 


74 


DESCRIPTIVE   GEOMETRY 


in  all  the  auxiliary  planes  and  will  pierce  the  plane  of  the  bases 
in  a  point  common  to  all  the  lines  which  the  auxiliary  planes 
will  cut  from  the  plane  of  the  bases. 


Fig.  49. — Intersection  of  pyramids. 

Construction.  The  line  through  the  vertices  Y^  and  Vg  pierces 
the  plane  of  the  bases  at  S.  Through  s  draw  st,  5^1,  st 2y  lines 
which  the  auxiliary  planes  cut  from  the  plane  of  the  bases.  The 
intersection  of  the  lines  which  the  auxiliary  planes  cut  from  the 
pyramids  are  points  on  the  required  line  of  intersection.  By 
using  only  the  planes  which  contain  edges  of  the  pyramids  and 
joining  in  order  the  points  thus  found  by  straight  lines,  the  line 
of  intersection  as  shown  in  Fig.  49  is  found. 

78.  To  find  the  line  of  intersection  of  the  surfaces  of  a  prism 
and  a  pyramid  when  the  edges  of  the  prism  are  parallel  to  the 
horizontal  plane,  by  the  use  of  an  auxiliary  plane  of  projection. 


INTERSECTIONS   OF  SURFACES 


75 


Let  the  prism  and  pyramid  be  given  as  in  Fig.  50. 
Analysis.     The  general  method  for  finding  the  line  of  inter- 
section of  a  prism  and  a  pyramid  will  solve  this  problem  but  a 


Fig.  50. — Intersection  of  prism  and  pyramid  hy  using  an  auxiliary 
plane  of  projection. 

much  simpler  method  is  to  use  an  auxiliary  view  taken  on  a 
plane  perpendicular  to  the  axis  of  the  prism.  In  this  view  the 
prism  is  represented  by  a  square  and  for  this  reason  the  points 
in  which  the  edges  of  the  pyramid  pierce  the  prism  are  found 
by  inspection.  To  find  the  points  in  which  an  edge  of  the  prism 
pierces  the  pyramid,  use  an  auxiliary  plane  containing  this  edge 
and  parallel  to  the  base  of  the  pyramid.  This  plane  cuts  lines 
from  the  faces  of  the  pyramid  which  intersect  the  edge  of  the 
prism  in  the  required  points. 

Construction.    In  the  auxiliary  view,  the  points  in  which  the 


76  DESCRIPTIVE   GEOMETRY 

lateral  edges  of  the  pyramid  pierce  the  faces  of  the  prism  are 
seen  to  be  iCg,  Wz,  Pz,  etc.  The  horizontal  views  x,  w,  p,  etc.,  of 
these  points  are  found  from  the  auxiliary  view  by  projection. 
The  vertical  views  are  then  found  from  the  horizontal  views. 
n^Uz  is  the  auxiliary  view  of  a  plane  which  is  parallel  to  the 
base  of  the  pyramid  and  contains  the  edge  NN^  of  the  prism. 
ut  and  uy  are  the  horizontal  views  of  the  lines  which  this  plane 
cuts  from  two  faces  of  the  pyramid  and  these  lines  cut  the  edge 
of  the  prism  in  the  required  points  T  and  Y.  Points  on  the 
other  edges  of  the  prism  can  be  found  in  the  same  way.  Join- 
ing the  points  thus  found  in  the  proper  order  gives  the  required 
line  of  intersection. 
79.  Problems. 

1.  The  lower  base  of  a  polyhedron  is  A( — 2f,  — 2S,  -^5),  B( — li,  —J, 
—5),  C(f,  —21,  —5),  D(— 2i,  —3 J,  —5).  The  upper  base  of  the  poly- 
hedron is  in  H,  one  corner  being  at  A  (i,  — 4,  0).  The  other  lateral 
edges  are  parallel  to  AA  .  Find  the  line  of  intersection  of  this  poly- 
hedron with  the  pyramid  O— BFG.  0(— 2i,  — |,  — i),  E(— -li,  — 4|, 
—5),  F(2i,  — li,  —5),  G(3i,  — 4i,  —5). 

2.  Find  the  line  of  intersection  of  a  cube  with  a  square  prism  when 
the  axis  of  the  prism  coincides  with  the  diagonal  of  the  cube.  Edge  of 
cube  2Y'',  side  of  base  of  prism  2i";  altitude  5".  Axis  of  prism  is  per- 
pendicular to  H,  the  plane  of  one  face  makes  15°  with  V.  The  H  pro- 
jection of  one  top  edge  of  the  cube  is  perpendicular  to  a  face  of  the 
prism. 

Develop  the  surface  of  the  cube  showing  the  line  of  intersection. 

3.  The  same  as  problem  2  except  that  the  prism  has  an  equilateral 
triangle  with  side  2i"  for  a  base. 

4.  The  same  as  problem  2  except  that  the  diagonal  of  the  cube  is  i" 
from  the  axis  of  the  prism. 

5.  Find  the  line  of  intersection  of  a  square  prism  and  a  right  square 
pyramid.  Prism  2''x2''xiY\  with  edges  parallel  to  H  and  30"*  with  V. 
The  plane  of  one  face  makes  30°  with  H  and  the  lowest  edge  is  J" 
above  the  base  of  the  pyramid.  The  base  of  the  pyramid  is  a  2i"  square 
resting  on  a  horizontal  plane,  side  of  base  60°  with  V.  The  axis  of  the 
pyramid  is  perpendicular  to  H  and  is  5''  long.  The  axes  of  the  prism 
and  the  pyramid  are  i"  apart. 

Note. — This  problem  can  be  multiplied  by  changing  the  number  of 
faces  of  the  prism  or  pyramid,  by  changing  the  distance  between  the 
axes,  by  changing  the  angle  which  the  plane  of  the  face  of  the  prism 
makes  with  H,  or  by  raising  or  lowering  the  prism. 


CHAPTER  IV 
CURVED  LINES  AND  SURFACES 


GENERATION  AND  CLASSIFICATION  OF  LINES 

80.  A  line  is  the  path  of  a  point  moving  according  to  some 
law. 

Lines  are  of  two  general  classes : 

I.  Straight  Lines  or  Right  Lines,  in  which  a  point  moves 
always  in  the  same  direction. 

n.  Curved  Lines,  in  which  a  point  moves  so  as  to  change  its 
direction  continually. 

Curved  lines  are  of  two  kinds : 

I.  Curves  of  Single  Curvature  or  Plane  Curves,  in  which  all 
positions  of  the  moving  point  lie  in  the  same  plane.  Examples, 
circle,  ellipse. 

II.  Curves  of  Double  Curvature,  in  which  four  positions  of 
the  moving  point  do  not  lie  in  the  same  plane  where  these  four 
points  are  taken  one  right  after  the  other  along  the  curve.  Ex- 
ample, the  edge  of  an  ordinary  screw  thread.  The  line  of  in- 
tersection of  two  curved  surfaces  is  usually  of  this  form. 


Lines 


OUTLINE 

I  Straight  or  Right  Lines 


II  Curved  Lines  -< 


Circle 
I  Curves  of  Single  Cur-     jj^^pge 

vature  or  Plane  Curves^  Parabola 


II  Curves    of   Double 
Curvature 


I  Hyperbola 
\  Helix 


78 


DESCRIPTIVE    GEOMETRY 


PROJECTIONS  OF  CURVES 

81.  The  cylinder  formed  by  the  horizontal  projecting  lines 
of  the  points  of  a  curve  is  the  horizontal  projecting  cylinder  of 
the  curve  and  its  intersection  with  the  horizontal  plane  is  the 
horizontal  projection  of  the  curve.  Likewise,  the  cylinder 
formed  by  the  vertical  projecting  lines  of  the  points  of  a  curve 
is  the  vertical  projecting  cylinder  of  the  curve  and  its  inter- 
section with  the  vertical  plane  is  the  vertical  projection  of  the 
curve,  Fig.  51. 

The  two  projections  of  a  curve 
will,  in  general,  determine  its  posi- 
tion in  space,  for  the  projecting  cyl- 
inders intersect  in  the  only  curve 
which  can  have  the  given  projec- 
tions. 

If  the  plane  of  a  curve  of  single 
curvature  is  perpendicular  to  the 
horizontal  plane,  the  horizontal  pro- 
jection of  the  curve  will  be  a  straight 
line. 

Likewise,  if  the  plane  of  the  curve  is  perpendicular  to  the 
vertical  plane,  the  vertical  projection  will  be  a  straight  line. 

If  the  plane  of  the  curve  is  perpendicular  to  the  ground  line, 
both  projections  will  be  straight  lines  perpendicular  to  the 
ground  line  and  the  curve  will  be  undetermined. 

If  the  plane  of  the  curve  is  parallel  to  the  horizontal  plane, 
the  horizontal  projection  will  be  equal  to  the  curve. 

Likewise,  if  the  plane  of  the  curve  is  parallel  to  the  vertical 
plane,  the  vertical  projection  will  be  equal  to  the  curve. 

The  projections  of  a  curve  of  double  curvature  are  always 
curved  lines. 


Fig.  51. — Projections  of 
a  curve. 


TANGENTS  AND  NORMALS   TO  LINES  79 


TANGENTS  AND  NORMALS  TO  LINES 

82.  If  in  a  secant  line  AB,  Fig.  52,  the  point  A  be  kept  fixed 
and  the  point  B  moved  along  the  curve  until  it  coincides  with  A, 
the  secant  AB  becomes  a  tangent  to  the  curve  at  the  point  A. 

Two  curves  are  said  to  be  tangent  to  each  other  at  a  point 
when  they  have  a  common  tangent  at  that  point. 

If  a  straight  line  is  tangent  to  a  plane 

A/  curve,  the  tangent  will  lie  in  the  plane 

Bi^,^^:::^^  of  the  curve.    This  is  evident  since  the 

Zl  C^  I  secant  through  the  point  of  tangency  is 

^^'^^^^^l.^*- \^        i^  t^®  plane  of  the  curve,  and  it  remains 
^         ^^     \       in  this  plane  as  it  moves  to  the  position 
Fig.  52. — Tangent  to  a        of  the  tangent. 

curve.  Two  straight  lines  tangent  to  each 

other  will  coincide.  In  this  case  the  secant  which  joins  two 
points  of  the  given  line  coincides  with  the  line  and  therefore 
the  tangent  must  coincide  with  the  given  line. 

83.  If  two  lines  are  tangent  to  each  other  in  space,  their  pro- 
jections on  the  same  plane  will  be  tangent.  Let  Fig.  52  repre- 
sent a  curve  in  space  with  its  secant  AB.  Let  these  lines  be 
projected  upon  any  plane.  Then  the  projections  of  the  points 
A  and  B  will  approach  each  other  as  the  points  A  and  B  in 
space  approach  each  other.  When  the  secant  AB  becomes  a 
tangent  in  space,  the  points  A  and  B  coincide  and  their  pro- 
jections will  also  coincide  in  the  only  point  common  to  the  pro- 
jections of  the  straight  line  and  curve.  The  projections  of  the 
lines  are  therefore  tangent  to  each  other  at  this  common  point. 

84.  Normals.  If  a  line  be  drawn  perpendicular  to  the  tan- 
gent at  the  point  of  tangency,  it  is  called  a  normal  to  the  curve. 
There  are  an  infinite  number  of  normals  to  a  curve  at  a  point 
on  the  curve.  However,  the  normal  to  a  plane  curve  is  under- 
stood to  be  the  one  lying  in  the  plane  of  the  curve  unless  other- 
wise stated. 


80  DESCRIPTIVE    GEOMETRY 

85.  Rectification  of  curves.  To  rectify  a  curve  means  to  find 
a  straight  line  equal  in  length  to  the  curve.  This  is  accom- 
plished approximately  by  dividing  the  curve  into  a  number  of 
small  arcs,  so  small  that  for  all  practical  purposes  the  chords 
of  these  arcs  may  be  taken  as  equal  in  length  to  the  arcs  them- 
selves. These  small  chords  are  then  laid  off  one  after  another 
along  a  straight  line.  The  part  of  the  line  thus  covered  is  the 
rectified  curve. 

CURVES  OF  SINGLE  CURVATURE 

86.  The  most  common  plane  curves  are  obtained  by  cutting  a 
right  circular  cone  by  a  plane.  These  curves  are  the  circle, 
ellipse,  parabola,  and  hyperbola.    Fig.  53  shows  the  position 


Fig.  53. — Plane  sections  of  a  cone  of  revolution. 

which  the  cutting  plane  must  take  with  reference  to  the  plane 
of  the  base  of  the  cone  in  order  to  cut  the  particular  curve  de- 
sired.   0  must  be  less  than  ©  and  oc  greater  than  ®. 

A  circle  is  the  path  of  a  point  moving  in  a  plane  so  that  its 
distance  from  a  given  point  is  constant. 

87.  An  ellipse  is  the  path  of  a  point  moving  in  a  plane  so  that 
the  sum  of  its  distances  from  two  fixed  points  is  constant. 

Construction.  Let  F  and  F^,  Fig.  54,  be  the  two  fixed  points 
and  let  the  sum  of  the  distances  from  these  points  to  any  point 
on  the  curve  be  equal  to  the  line  AB.  Select  any  point,  as  0, 
on  the  line  AB.     With  F  as  center  and  AO  as  radius,  strike  an 


CURVES  OP  SINGLE  CURVATURE 


81 


arc;  also  with  Fi  as  center  and  BO  as  radius,  strike  an  arc 
cutting  the  first  arc  in  the  points  P  and  Q.  These  are  points  on 
the  curve  since  FP+FiP=AB  and  FQ4-FiQ=AB.  Selecting 
any  other  point  on  AB,  as  Oi,  and  going  through  the  construc- 
tion as  before,  two  other  points  Pi  and  Qi  are  located.  By  con- 
tinuing this  process  enough  points  can  be  located  so  that  the 
curve  can  be  drawn. 


--JM 


Fig,  54. — Ellipse  and  tangents. 


The  points  F  and  Fj  are  foci  of  the  ellipse.  AB  is  the  major 
axis.  The  minor  axis  CD  is  perpendicular  to  the  major  axis  at 
its  middle  point.  The  intersection  of  the  axes  is  the  center  of 
the  ellipse.  Lines  drawn  from  any  point  on  the  curve,  as  P,  to 
the  foci  F  and  F^  are  called  focal  radii. 

If  the  axes  of  an  ellipse  are  given,  the  foci  can  be  found  by 
taking  one-half  the  major  axis  as  a  radius  and  with  the  end  of 
the  minor  axis  as  a  center,  cut  the  major  axis  in  two  points 
which  are  the  foci. 

88.  To  draw  a  tangent  to  the  ellipse  at  a  point  on  the  curve. 
Let  P,  Fig.  54,  be  the  point  on  the  curve.  Draw  the  focal  radii 
PF  and  PF^.     The  line  PT  which  bisects  the  angle  F^PE  is  the 


82 


DESCRIPTIVE   GEOMETRY 


required  tangent.  (Wood^s  Co-ordinate  Geometry,  Art.  108A.) 
This  method  applies  to  all  conies. 

To  draw  a  tangent  to  the  ellipse  from  a  point  without  the 
curve.  Let  S,  Fig.  54,  be  the  point  without  the  curve.  With 
Fi  as  center  and  AB  as  radius,  strike  an  arc.  With  S  as  center 
and  SF  as  radius,  strike  an  arc  cutting  the  first  arc  in  the  points 
M  and  N.  F^M  and  F^N  intersect  the  curve  in  the  points  of 
tangency  Ti  and  Tg.  ST^  and  ST2  are  the  required  tangents. 
(Wood's  Co-ordinate  Geometry,  Art.  108B.)  This  method  ap- 
plies to  all  conies. 

89.  A  parabola  is  the  path  of  a  point  moving  in  a  plane  so 
that  its  distance  from  a  given  point  is  always  equal  to  its  dis- 
tance from  a  given  straight  line. 


Fig.  55. — Parabola  and  tangents. 

Construction.  Let  F,  Fig.  55,  be  the  given  point  and  AB  the 
given  straight  line.  Draw  a  line  CD  from  F  perpendicular  to 
AB.  Through  any  point,  as  0,  on  CD,  draw  a  line  parallel  with 
AB.  With  F  as  center  and  CO  as  radius  strike  an  arc  cutting 
this  line  in  the  two  points  P  and  Q.  P  and  Q  are  points  on  the 
parabola,  since  FP  and  FQ  are  each  equal  to  the  distance  of  P 


CURVES  OF  SINGLE  CURVATURE  83 

and  Q  from  AB.  Selecting  some  other  point  on  the  axis  and 
going  through  a  similar  construction,  two  other  points  of  the 
curve  are  located.  By  continuing  this  process,  enough  points 
can  be  located  so  that  the  curve  can  be  drawn. 

The  point  F  is  the  focus,  the  line  AB  the  directrix,  and  the 
line  CD  the  axis  of  the  parabola. 

90.  To  draw  a  tangent  to  the  parabola  at  a  point  on  the 
curve.  Let  P^  Fig.  55,  be  the  point  on  the  curve.  Draw  the 
focal  radii  FPi  and  FjF^,  (To  make  this  construction  similar 
to  that  for  the  tangent  to  the  ellipse,  it  is  necessary  to  consider 
one  focus  Fi  of  the  parabola  at  an  infinite  distance  on  the  axis.) 
The  line  PjT  which  bisects  the  angle  FPjE  is  the  required  tan- 
gent. 

To  draw  a  tangent  to  the  parabola  from  a  point  without  the 
curve.  Let  S,  Fig.  55,  be  the  point  without  the  curve.  With  S 
as  center  and  SF  as  radius,  strike  an  arc  cutting  the  directrix 
in  the  points  M  and  N.  MF^  and  NF^,  parallel  with  the  axis 
CD  of  the  parabola,  cut  the  curve  in  the  points  of  tangency  Ti 
and  Tg.     ST^  and  STg  are  the  required  tangents. 

91.  A  hyperbola  is  the  path  of  a  point  moving  in  a  plane  so 
that  the  difference  of  its  distances  from  two  fixed  points  is  con- 
stant. 

Construction.  Let  F  and  Fi,  Fig.  56,  be  the  two  fixed  points 
and  let  the  difference  of  the  distances  from  these  points  to  any 
point  on  the  curve  be  equal  to  the  line  AB.  Select  any  point, 
as  0,  on  the  line  AB  extended.  With  F  as  center  and  AO  as 
radius,  strike  an  arc ;  also  with  F^  as  center  and  BO  as  radius, 
strike  an  arc  cutting  the  first  arc  in  the  points  P  and  Q.  These 
are  points  on  the  curve  since  FP — ^FiP=AB  and  FQ — ^FiQ=AB. 
(If  Fi  be  taken  as  center  and  AO  as  radius  and  F  as  center  and 
BO  as  radius,  the  points  located  will  be  on  the  other  branch  of 
the  hyperbola.)  Selecting  any  other  point  on  AB,  as  Oi,  and 
going  through  the  construction  as  before,  two  other  points  Pi 
and  Qi  are  located.  By  continuing  this  process  enough  points 
can  be  located  so  that  the  curve  can  be  drawn. 


84 


DESCRIPTIVE   GEOMETRY 


The  points  F  and  Fj  are  the  foci  of  the  hyperbola.  The  point 
C,  midway  between  the  foci  F  and  Fi  is  the  center  of  the  curve. 
The  line  AB  is  the  transverse  axis.  The  line  XY,  perpendicular 
to  AB  at  its  middle  point,  is  the  indefinite  conjugate  axis  of  the 
curve. 


Fig.  56. — Hyperbola  and  tangents. 


92.  To  draw  a  tangent  to  the  hyperbola  at  a  point  on  the 
curve.  Let  P^,  Fig.  56,  be  the  point  on  the  curve.  Draw  the 
focal  radii  FP^  and  F^P^.  The  line  PjT  which  bisects  the  angle 
FjPiF  is  the  required  tangent. 

To  draw  a  tangent  to  the  hyperbola  from  a  point  without  the 
curve.  Let  S,  Fig.  56,  be  the  point  without  the  curve.  With 
Fi  as  center  and  AB  as  radius,  strike  an  arc.  With  S  as  center 
and  SF  as  radius,  strike  an  arc  cutting  the  first  arc  in  the  points 
M  and  N.  F^M  and  F^N  intersect  the  curve  in  the  points  of 
tangency.  ST^  is  one  of  the  tangents.  The  other  point  of  tan- 
gency,  where  NFj  intersects  the  curve,  is  without  the  limits  of 
the  drawing. 


CURVES  OF  DOUBLE  CURVATURE 


85 


CURVES  OF  DOUBLE  CURVATURE 

93.  An  ordinary  helix  is  the  path  of  a  point  moving  on  the 
surface  of  a  cylinder  of  revolution  so  as  to  intersect  its  elements 
at  a  constant  acute  angle. 

The  axis  of  the  cylinder  is  the  axis  of  the  helix. 
Construction.    Let  MN,  Fig.  57,  be  the  axis  of  the  helix  and  P 
the  generating  point. 

Suppose  that  for  one  complete  turn  around  the  axis  the  gen- 
erating point  moves  through  a  vertical  distance  m'n\  This  dis- 
tance is  the  pitch  of  the  helix. 

Since  the  curve  is  on  the  surface  of  a 
cylinder  of  revolution,  its  projection  on 
a  plane  perpendicular  to  the  axis  will  be 
the  circle  pcfg. 

To  draw  the  vertical  projection  of  the 
helix,  divide  the  circle  pcfg  into  any 
number  of  equal  parts,  as  twelve,  and 
m'7^'  into  the  same  number  of  equal 
parts.  Draw  horizontal  lines  through 
the  points  of  division  of  m'n\  Since  the 
motions  of  the  point  around  and  along 
the  axis  are  both  uniform,  the  point  in 
making  one-twelfth  of  a  complete  turn 
around  the  axis  will  rise  one-twelfth  of 
the  distance  mV.  a  is  the  horizontal 
and  a'  the  vertical  projection  of  the 
point  after  making  one-twelfth  of  a 
turn.  In  the  same  manner,  the  points 
h\  c\  d',  etc.,  are  found.  The  vertical  projection  of  the  curve  is 
drawn  through  the  vertical  projections  of  these  points. 

Since  the  curve  cuts  all  the  elements  of  the  cylinder  at  the 
same  angle,  it  is  evident  that  the  helix  will  become  a  straight 
line  if  the  cylinder  is  opened  along  an  element  and  the  surface 
rolled  out  into  a  plane.  From  this  position  it  is  seen,  that  the 
hypothenuse  of  a  right  triangle  will  form  a  helix  if  the  altitude 


P' 

Fig.  57. — Helix  and 
tangent. 


86  DESCRIPTIVE   GEOMETRY" 

of  the  triangle  coincides  with  an  element  of  a  right  circular 
cylinder  and  the  base  of  the  triangle  is  wound  around  the  base 
of  the  cylinder. 

94.  To  construct  a  tangent  to  a  helix  at  a  point  on  the  curve. 
Let  B,  Fig.  57,  be  the  point  at  which  the  tangent  is  to  be  drawn. 
Let  the  triangle  referred  to  in  the  above  paragraph  as  being 
wound  around  the  cylinder,  be  unrolled  as  far  as  the  point  of 
tangency  B.  The  hypothenuse  of  the  triangle  in  this  position 
is  tangent  to  the  curve,  since  it  touches  the  curve  at  the  point  B 
and  makes  the  same  angle  with  the  horizontal  plane  as  the 
curve.  The  base  ht  of.  the  triangle  is  the  horizontal  projection 
of  the  tangent,  and  is  equal  in  length  to  the  arc  hp  of  the  circle 
pcfg*  Therefore,  to  get  the  projections  of  the  tangent,  draw  its 
horizontal  projection  ht  tangent  to  the  circle  pcfg  at  the  point  h. 
On  this  line  lay  off  from  h  the  true  length  of  the  arc  hap  of  the 
circle.  This  will  give  the  point  t  where  the  tangent  pierces  the 
plane  of  the  base.  The  vertical  projection  of  this  piercing  point 
is  at  f  and  this  joined  with  h'  gives  the  vertical  projection  h'f 
of  the  tangent. 


GENERATION  AND  CLASSIFICATION  OF  SURFACES 

95.  A  surface  is  the  path  of  a  line  moving  according  to  some 
law. 

The  moving  line  is  the  generatrix,  and  its  different  positions 
are  the  elements  of  the  surface. 

Surfaces  are  of  two  general  classes : 

I.  Ruled  surfaces,  which  can  be  generated  by  straight  lines. 
Examples,  cylinder,  cone,  helicoid. 

II.  Double  curved  surfaces,  which  can  only  be  generated  by 
curved  lines.     Example,  sphere,  ellipsoid. 

Ruled  surfaces  are  of  three  kinds : 

I.  Plane  surfaces. 

II.  Single  curved  surfaces,  which  can  be  rolled  into  a  plane 
without  undergoing  distortion.     Examples,  cylinder,  cone. 


GENERATION  AND   CLASSIFICATION  OP  SURFACES 


87 


III.  Warped  surfaces,  which  cannot  be  rolled  into  a  plane 
without  undergoing  distortion.  Example,  the  surface  of  a 
screw  thread. 

Single  curved  surfaces  are  of  three  kinds : 

I.  Cylinders,  in  which  the  rectilinear  elements  are  parallel. 

II.  Cones,  in  which  the  rectilinear  elements  intersect  in  a 
point. 

III.  Convolutes,  in  which  the  rectilinear  elements  are  tan* 
gent  to  a  curve  of  double  curvature.  Example,  the  surface  hav- 
ing tangents  to  a  helix  for  its  elements. 


OUTLINE 

I  Plane 

{I  Cylinder 
II  Cone 
III  Convolute 


Surfaces 


I  Euled 


III  Warped  < 


Helicbid 

Hyperbolic  Paraboloid 
Conoid 
Cylindroid 

Hyperboloid  of  Revo- 
lution of  one  Sheet 


II  Double  curved  < 


Sphere 

Torus  or  Anchor  ring 
Ellipsoid  of  Revolution 
Paraboloid  of  Revolution 
Hyperboloid  of  Revolution  of 
two  Sheets 


SURFACES  OF  REVOLUTION 

96.  A  surface  of  revolution  is  the  path  of  a  line  revolving 
about  a  straight  line  as  an  axis. 

It  is  evident  that  the  intersection  of  this  surface  with  a  plane 
perpendicular  to  the  axis  is  a  circle.    A  surface  of  revolution: 


88 


DESCRIPTIVE   GEOMETRY 


may  be  generated  by  a  circle  having  its  center  moving  along  a 
straight  line,  its  different  positions  in  parallel  planes,  and  its 
radius  changing  according  to  a  given  law. 

The  intersection  of  the  surface  with  a  plane  containing  the 
axis  is  a  meridian  line,  and  the  plane  is  a  meridian  plane.  All 
meridian  lines  of  the  same  surface  are  identical.  Any  surface 
of  revolution  may  be  generated  by  revolving  its  meridian  line 
about  its  axis. 

OUTLINE 


Surfaces  of 
Revolution 


Ruled 


^.     -  -,    (  Right  circular  cylinder 

Single  curved  -^  -d-  i.    •      i 

*  I  Right  circular  cone 

\  TKT        A    ^  Hyperboloid  of  Revolution 
Warped   |     of  one  Sheet 


Double  curved  ^ 


'  Sphere 
Torus 
ElKpsoid  of  Revolu-  (  Prolate 

tion  ( Oblate 

Paraboloid  of  Revolution 
Hyperboloid  of  Revolution  of 

two  Sheets 


97.  If  two  surfaces  of  revolution  have  a  common  axis,  what 
will  be  their  line  of  intersection? 

What  must  be  the  relative  position  of  the  axis  and  the  gen- 
eratrix to  form  a  cylinder  of  revolution?  A  cone  of  revolution? 
A  hyperboloid  of  revolution  of  one  sheet  ? 

The  cylinder  and  cone  are  the  only  single  curved  surfaces  of 
revolution. 

The  hyperboloid  of  revolution  of  one  sheet  is  the  only  warped 
surface  of  revolution. 


TANGENT  PLANES  TO  SURFACES 


89 


TANGENT  PLANES  TO  SURFACES.    NORMAL  LINES  AND  PLANES 

98.  Let  ADi  and  ADg,  Fig.  58,  be  any  two  intersecting  curves 
on  a  surface,  and  BCD  a  curve  which,  if  moved  along  the  curves 
ADi  and  ADg  will  generate  the  given  surface.  The  curve  BCD 
may  vary  in  form  as  it  moves.  The  secants  AB,  BC,  and  AC  lie 
in  one  plane.  As  the  curve  BCD  moves  toward  A,  the  points  of 
intersection  B  and  C  will  travel  along  the  curves  AD^  and  ADg 
until  the  secants  AB  and  AC  become  the  tangents  AB^  and  ACg 
at  the  point  A.  The  curve  ADg  is  the  position  of  the  moving 
curve  BCD  when  the  points  B  and  C  reach  A.  In  this  position, 
the  secant  BC  becomes  the  tangent  B3C3.  The  tangents  ABj, 
AC2,  and  B3C3  all  lie  in  one  plane  which  is  called  the  tangent 
plane  to  the  surface  at  the  point  A.  The  point  A  is  the  point 
of  contact. 

Since  AD^  and  AD2  are  any 
curves  of  the  surface,  it  follows 
that,  in  general,  the  tangent  at 
A  to  every  curve  of  the  surface 
through  this  point  will  lie  in  the 
tangent  plane.  Therefore,  the 
tangent  plane  will  contain  all 
straight  lines  tangent  to  lines 
of  the  surface  at  the  point  of 
contact. 

If  any  plane  be  passed  through  the  point  of  contact,  it  will 
cut  a  straight  line  from  the  tangent  plane  and  a  line  from  the 
surface,  and  these  lines  will  be  tangent  to  each  other. 

Since  two  intersecting  lines  will  determine  a  plane,  it  follows 
that  a  plane  tangent  to  a  surface  at  a  given  point  is  determined 
by  two  straight  lines  tangent  at  this  point  to  two  lines  of  the 
surface. 

In  any  ruled  surface,  the  rectilinear  element  through  the 
point  of  contact  lies  in  the  tangent  plane;  for  the  rectilinear 
element  and  its  tangent  coincide  (Art.  82). 


Fig.  58. — Tangent  plane  to  a  sur- 
face. 


90  DESCRIPTIVE   GEOMETRY 

99.  A  tangent  plane  to  a  single  curved  surface  is  tangent  all 
along  a  rectilinear  element.  In  Fig.  59,  let  BD  be  the  curve  of 
the  base  of  any  single  curved  surface  as  a  cylinder  and  AB  and 
AjBi  any  two  rectilinear  elements  of  the  surface.  From  any 
point  A  of  one  element  draw  on  the  surface  a  curve  AD^  which 
cuts  the  other  element  at  A^.  The  chords  AA^  and  BB^  Ue  in 
the  same  plane  and  will  intersect  at  some  point  as  Si.  Now  if 
the  plane  ABB^Ai  be  revolved  about  AB  as  an  axis,  the  point  A^ 
will  approach  A  along  the  curve  AD^  and  B^  will  approach  B 
along  the  curve  BD.  When  the  point  A^  reaches  A,  the  secant 
ASi  becomes  the  tangent  AS  to  the  curve  AJD^,  and  at  the  same 
time  Bi  reaches  B,  the  secant  S^B  becoming  the  tangent  SB  to 
the  curve  BD.  The  plane  SAB  is  tangent  to  the  cylinder  at  the 
point  B  (Art.  98),  since  it  contains  the  tangent  SB  to  the  curve 
BD,  and  also  the  rectilinear  element  AB.  This  plane  is  also 
tangent  to  the  cylinder  at  A,  containing  the  tangent  SA  to  the 
curve  ADi  and  the  element  AB.  Since  A  was  taken  as  any 
point  on  the  element  AB,  the  plane  SAB  must  be  tangent  to  the 
surface  at  every  point  of  this  element. 

The  same  demonstration  ap- 
plies to  the  tangent  plane  to  a 
cone. 

In  the  case  of  the  convolute, 
the  element  A^B^  is  a  curve  in- 
stead of  a  straight  line.  The 
curve  will  change  its  form  as 
the  secant  plane  revolves,  finally 
becoming  a  straight  line  coin- 
FiQ.  59.— Tangent  plane  to  single  ciding  with  the  axis  AB.  Other- 
curved  surface.  ^ise   the   demonstration   given 

above  also  applies  to  the  convolute. 

Since  the  plane  is  tangent  all  along  the  element,  the  intersec- 
tion of  the  tangent  plane  with  the  plane  of  the  base  will  be  a 
straight  line  tangent  to  the  curve  which  represents  the  base 
(Art.  98). 


TANGENT  PLANES  TO  SURFACES  91 

100.  Tangent  planes  to  warped  surfaces.  Since  warped  sur- 
faces have  straight  line  elements,  the  tangent  plane  to  the  sur- 
face at  any  point  will  contain  the  rectilinear  element  through 
that  point  (Art.  98).  If  any  curve  of  the  surface  be  drawn 
through  the  point  of  contact,  the  straight  line  tangent  to  the 
curve  at  this  point  and  the  rectilinear  element  through  the  point 
will  determine  the  tangent  plane  (Art.  98). 

If  the  warped  surface  is  of  such  a  form  that  there  are  two 
rectilinear  elements  through  each  point  of  the  surface,  then  the 
tangent  plane  will  be  determined  by  the  two  elements  through 
the  point  of  contact. 

Although  the  tangent  plane  to  a  warped  surface  contains  a 
rectilinear  element  of  the  surface,  it  is,  in  general,  tangent  at 
only  one  point  of  this  element.  The  tangent  plane  usually  in- 
tersects the  surface.  These  principles  will  be  better  under- 
stood when  the  tangent  planes  to  some  of  the  surfaces  are 
drawn. 

101.  Tangent  planes  to  double  curved  surfaces.  Through  the 
point  on  the  surface  at  which  the  tangent  plane  is  to  be  passed, 
draw  any  two  curves  of  the  surface.  The  straight  lines  tan- 
gent to  these  curves  at  their  point  of  intersection  will  determine 
the  tangent  plane.  If  care  is  taken  to  select  simple  curves  of 
the  surface  in  simple  positions  with  reference  to  the  planes  of 
projection,  the  tangents  to  these  can  be  easily  drawn. 

In  the  case  of  a  double  curved  surface  of  revolution,  the 
simplest  curves  are  usually  the  meridian  curve  and  a  circle 
which  lies  in  a  plane  perpendicular  to  the  axis  of  the  surface. 

102.  Normals.  A  straight  line  perpendicular  to  the  tangent 
plane  at  the  point  of  contact  is  the  normal  to  the  surface  at  that 
point.  Any  plane  containing  a  normal  line  is  a  normal  plane 
to  the  surface. 


92  DESCRIPTIVE   GEOMETRY 

SINGLE  CURVED  SURFACES 

CYLINDERS 

103.  A  cylinder  is  the  path  of  a  straight  line  moving  along  a 
curve  and  remaining  parallel  to  another  straight  line. 

The  curve  is  the  directrix  and  the  moving  line  is  the  recti- 
linear generatrix. 

The  directrix  can  be  a  closed  curve  as  the  circle  or  ellipse  or 
a  curve  which  is  not  closed  as  the  hyperbola  or  helix. 

The  intersection  of  the  cylinder  with  a  horizontal  plane  is 
usually  taken  as  the  base.  Any  plane  section  may  be  consid- 
ered to  be  the  base.  If  this  base  has  a  center,  the  straight  line 
through  it  and  parallel  to  the  rectilinear  elements  is  the  axis  of 
the  cylinder. 

A  right  cylinder  is  one  in  which  the  rectilinear  elements  are 
perpendicular  to  the  plane  of  the  base. 

A  section  taken  perpendicular  to  the  elements  of  any  cylinder 
is  a  riifht  section.  If  a  cylinder  is  intersected  by  a  plane  par- 
allel to  the  rectilinear  generatrix  it  will  cut  rectilinear  ele- 
ments from  the  surface. 

104.  To  represent  the  surface.  A  cylinder  is  usually  repre- 
sented by  the  projections  of  its  base  and  its  extreme  elements. 

In  Fig.  60,  AECG  is  the  base  and  EF  the  rectilinear  genera- 
trix. The  extreme  elements  in  the  horizontal  view  are  tangent 
to  the  base  and  parallel  to  the  horizontal  projection  of  the  rec- 
tilinear generatrix.  The  extreme  elements  in  the  vertical  view 
are  drawn  through  the  extreme  points  of  the  vertical  projec- 
tion of  the  base  and  parallel  to  the  vertical  projection  of  the 
rectilinear  generatrix. 

Since  the  surface  is  indefinite  in  extent,  it  can  be  represented 
with  the  upper  end  broken  off  unless  some  definite  portion  of 
the  surface  is  taken  for  a  particular  purpose. 

105.  To  represent  a  rectilinear  element  of  the  surface. 
Through  any  point  M  of  the  base  draw  a  straight  line  MN  par- 


CYLINDERS 


93 


allel  to  the  rectilinear  generatrix ;  this  will  be  an  element  of  the 
surface. 

To  represent  a  point  of  the  surface.  Locate  a  rectilinear 
element,  as  MN,  and  then  take  any  point  P  of  this  element. 

106.  To  represent  a  plane  which  contains  a  given  point  and  is 
tangent  to  a  cylinder. 

The  tangent  plane  to  a  cylinder  is  tangent  all  along  a  rectilinear 
element. 


Fig.  60. — Cylinder. 

The  intersection  of  the  tangent  plane  with  the  plane  of  the  hose 
is  ta/ngent  to  the  base. 

Let  the  cylinder  be  given  as  in  Fig.  61,  and  let  P  be  the  given 
point. 

Since  the  tangent  plane  must  contain  a  rectilinear  element  of 
the  cylinder,  a  straight  line  through  P  and  parallel  to  the  rec- 
tilinear elements  will  lie  in  the  tangent  plane  and  will  pierce 
the  plane  of  the  base  at  M.  A  tangent  to  the  base  of  the  cylin- 
der through  the  point  M  is  a  horizontal  of  the  required  tangent 
plane.    An  element  of  the  cylinder  through  the  point  of  tan- 


94 


DESCRIPTIVE    GEOMETRY 


gency  0  will  be  the  element  of  contact.     This  element,  the  hori- 
zontal OM,  and  the  line  PM  will  determine  the  tangent  plane. 

There  can  be  as  many  tangent  planes  to  the  cylinder  through 
the  given  point  as  there  can  be  lines  drawn  tangent  to  the  base 
of  the  cylinder  through  the  point  where  the  auxiliary  line 
pierces  the  plane  of  the  base. 


Fig.  61. — Tangent  plane  to  cylinder. 


107.  Problems. 

1.  A  right  circular  -cylinder  and  a  point  are  given  by  their  projec- 
tions. Represent  a  plane  which  contains  the  point  and  is  tangent  to 
the  cylinder. 

2.  Represent  a  plane  which  is  tangent  to  a  cylinder  at  a  given  point 
of  the  surface. 

3.  Represent  a  plane  which  is  tangent  to  a  cylinder  and  parallel  to  a 
given  straight  line.  (A  plane  containing  the  given  line  and  a  line 
parallel  to  the  elements  of  the  cylinder  will  be  parallel  to  the  required 
tangent  plane.) 

4.  Given  the  projections  of  a  point  and  a  right  circular  cylinder 
which  has  its  axis  parallel  to  the  ground  line.  Represent  a  plane  which 
contains  the  point  and  is  tangent  to  the  cylinder. 

5.  Represent  a  plane  which  makes  45°  with  H  and  is  tangent  to  a 
right  circular  cylinder.  The  axis  of  the  cylinder  is  parallel  to  H  and 
makes  30°  with  V. 


CONES 


95 


CONES 

108.  A  cone  is  the  path  of  a  straight  line  moving  along  a 
curve,  and  passing  through  a  fixed  point.  If  the  directrix  is  a 
plane  curve,  the  fixed  point  does  not  lie  in  the  plane  of  the 
curve. 

The  curve  is  the  directrix,  the  fixed  point  the  vertex  and  the 
moving  line  the  rectilinear  generatrix  of  the  cone. 


m'        g' 
Fig.  ^2.— Cone. 


The  directrix  can  be  a  closed  curve  as  a  circle  or  an  ellipse 
or  a  curve  which  is  not  closed  as  a  hyperbola  or  a  helix. 

The  generatrix  being  indefinite  in  length,  will  generate  a  sur- 
face of  two  parts  which  are  on  opposite  sides  of  the  vertex  and 
which  are  called  the  upper  and  lower  sheets. 

The  intersection  of  the  cone  with  a  horizontal  plane  is  usu- 
ally taken  as  the  base,  but  any  plane  section  of  the  cone  may  be 
considered  the  base. 


96  DESCRIPTIVE   GEOMETRY 

A  right  cone  is  one  in  which  all  the  rectilinear  elements  make 
the  same  angle  with  a  straight  line  through  the  vertex  called 
the  axis  of  the  cone. 

Any  plane  passing  through  the  vertex  and  intersecting  the 
cone  will  cut  rectilinear  elements  from  the  surface. 

109.  To  represent  the  surface.  A  cone  is  usually  represented 
by  the  projections  of  its  base,  vertex  and  extreme  elements. 

In  Fig.  62,  AECG  is  the  base  and  V  the  vertex  of  the  cone. 
The  extreme  elements  in  the  horizontal  view  are  tangent  to  the 
base  and  pass  through  the  horizontal  projection  of  the  vertex. 
The  extreme  elements  in  the  vertical  view  are  drawn  through 
the  extreme  points  of  the  vertical  projection  of  the  base  and 
through  the  vertical  projection  of  the  vertex. 

110.  To  represent  a  rectilinear  element  of  the  surface. 
Through  any  point  M  of  the  base,  draw  a  straight  line  through 
the  vertex  V ;  this  will  be  an  element  of  the  surface. 

To  represent  a  point  of  the  surface.  Locate  a  rectilinear  ele- 
ment, as  MV,  and  take  any  point  P  of  this  element. 

111.  To  represent  a  plane  which  contains  a  given  point  and  is 
tangent  to  a  cone. 

The  tcmgent  plane  to  a  cone  is  tangent  all  along  a  rectilinear 
element. 

The  intersection  of  the  tangent  plane  with  the  plane  of  the  base 
is  tangent  to  the  base. 

Let  the  cone  be  given  as  in  Fig.  63,  and  let  P  be  the  given 
point. 

Since  the  required  plane  must  contain  a  rectilinear  element 
of  the  cone,  and  therefore  the  vertex,  a  straight  line  joining  the 
vertex  with  the  given  point  P  will  lie  in  the  tangent  plane  and 
pierce  the  plane  of  the  base  at  M.  A  line  through  the  point  M 
and  tangent  to  the  base  of  the  cone  is  a  horizontal  of  the  re- 
quired tangent  plane.  An  element  through  the  point  of  tan- 
gency  0  is  the  element  of  contact.  This  element,  the  horizon- 
tal and  the  line  YP  all  lie  in  the  required  tangent  plane. 


CONES 


97 


There  are  as  many  solutions  to  the  problem  as  there  are  tan- 
gents to  the  base  of  the  cone  from  the  point  in  which  the  auxil- 
iary line  pierces  the  plane  of  the  base.  The  problem  cannot  be 
solved  when  the  given  point  is  inside  the  cone. 


m'  o' 

Fig.  63. — Tangent  plane  to  cone. 


112.  Problems. 

1.  A  right  cone  and  a  point  are  given  by  their  projections.  Repre- 
sent a  plane  which  contains  the  point  and  is  tangent  to  the  cone. 

2.  A  right  cone  with  its  base  in  a  profile  plane  and  a  point  are  given 
by  their  projections.  Represent  a  plane  which  contains  the  point  and 
is  tangent  to  the  cone. 

3.  A  right  cone  with  its  axis  parallel  to  H  and  oblique  to  V  and  a 
point  are  given  by  their  projections.  Represent  a  plane  which  contains 
the  point  and  is  tangent  to  the  cone. 

4.  Represent  a  plane  which  is  tangent  to  a  given  oblique  cone  at  a 
point  on  the  surface. 

5.  Represent  a  plane  which  is  tangent  to  a  given  cone  and  parallel 
to  a  given  straight  line.  (A  line  through  the  vertex  of  the  cone  and 
parallel  to  the  given  line  will  lie  in  the  required  tangent  plane.) 


98 


DESCRIPTIVE   GEOMETRY 


CONVOLUTES 

113.  A  convolute  is  the  path  of  a  straight  line  moving  along 
and  remaining  tangent  to  a  curve  of  double  curvature  such  as 
the  helix.     The  surface,  Fig.  64,  is  represented  by  drawing  the 


Fig.  64. — Convolute. 

projections  of  the  curvilinear  directrix,  an  occasional  element 
and  the  intersection  of  the  surface  with  a  horizontal  plane.  If 
the  curvilinear  directrix  is  a  helix  with  axis  perpendicular  to 
the  horizontal  plane,  the  base  of  the  surface  will  be  the  in- 
volute of  the  circle  which  represents  the  horizontal  projection 
of  the  helix. 

This  surface  is  of  very  little  practical  importance,  so  no  fur- 
ther discussion  of  it  will  be  given.  For  a  complete  discussion 
of  it,  see  Salmon's  Geometry  of  Three  Dimensions,  Art.  349. 


WABPED  SURFACES  99 


WARPED  SURFACES 

114.  There  is  a  great  variety  of  warped  surfaces,  differing 
from  each  other  in  their  mode  of  generation  and  properties. 
Only  a  few  of  the  more  common  ones  will  be  discussed. 

There  are  several  kinds  of  warped  surfaces  formed  by  mov- 
ing a  straight  line  to  touch  two  other  lines,  straight  or  curved, 
and  keeping  it  parallel  to  a  given  plane,  called  a  plane  directer. 

Warped  surfaces  are  usually  represented  by  drawing  the 
projections  of  one  or  more  curves  of  the  surface  and  a  few  of 
the  rectilinear  elements.  The  projections  of  the  directrices  are 
often  given. 

HYPERBOLOIDS  OP  REVOLUTION  OF  ONE  SHEET 

115.  A  hyperboloid  of  revolution  of  one  sheet  is  the  path  of  a 
straight  line  revolving  about  another  straight  line  as  an  axis, 
the  generatrix  and  axis  not  being  in  the  same  plane. 

From  the  nature  of  the  motion  it  is  evident  that  no  two  posi- 
tions of  the  generatrix  can  be  brought  into  one  plane  without 
distortion,  therefore  the  surface  is  warped  (Art.  95). 

This  is  the  only  warped  surface  of  revolution. 

116.  To  represent  the  surface.  In  Fig.  65,  MN  is  the  axis  and 
AB  the  initial  position  of  the  generatrix.  Each  point  of  the 
generatrix  AB  describes  a  horizontal  circle.  The  point  R,  near- 
est the  axis,  describes  a  circle  of  radius  nr,  called  the  circle  of 
the  gorge.  The  point  A  describes  a  circle  on  a  horizontal  plane 
usually  considered  the  base  of  the  surface.  The  horizontal 
projection  of  the  base  and  of  the  circle  of  the  gorge  form  the 
horizontal  view  of  the  surface. 


100 


DESCRIPTIVE   GEOMETRY 


In  each  position  of  the  genera- 
trix AB  there  is  a  radius  of  the 
circle  of  the  gorge  perpendicular 
to  it.  Since  the  radius  is  parallel 
to  H,  its  horizontal  projection  will 
be  perpendicular  to  the  horizon- 
tal projection  of  the  generatrix. 
Hence  the  horizontal  projections 
of  the  elements  of  the  surface  are 
tangent  to  the  horizontal  projec- 
tion of  the  circle  of  the  gorge. 
The  vertical  projection  of  the  cir- 
cle of  the  gorge  is  a  horizontal  line 
through  /  and  the  vertical  pro- 
jection of  the  base  is  a  horizontal 
line  through  a'.  Therefore,  to  rep- 
resent an  element  of  the  surface, 
draw  its  horizontal  projection  cd 
tangent  to  the  horizontal  projec- 
tion of  the  circle  of  the  gorge.  The 
vertical  projection  of  C  is  c'  and  of 
the  point  of  tangency  E  is  e'. 
Hence  c'e'd'  is  the  vertical  projec- 
tion of  the  element. 
The  vertical  projection  of  the  surface  is  the  hyperbola  which 
is  the  envelope  of  the  vertical  projections  of  the  rectilinear 
elements  of  the  surface.  Definite  points  on  this  hyperbola  are 
found  by  finding  the  points  in  which  the  rectilinear  elements 
pierce  the  meridian  plane  which  is  parallel  to  V. 

To  represent  a  point  of  the  surface.  Locate  an  element  CD 
and  then  take  any  point  P  of  this  element. 

117.  In  Fig.  65,  draw  a  line  XY  through  the  point  R  making 
the  same  angle  with  H  and  having  the  same  horizontal  projec- 
tion as  the  generatrix  AB.  For  each  point  of  XY  there  is  a 
corresponding  point  of  AB  which  is  the  same  distance  from  H 
and  also  the  same  distance  from  MN.    As  AB  and  XY  revolve 


FiQ.  65. — Hyperholoid  of  rev- 
olution of  one  sheet  and  tan- 
gent plane. 


WARPED  SURFACES  101 

about  MN  as  an  axis,  these  two  points  will  generate  the  same 
circle  and  the  two  lines  will  therefore  generate  the  same  sur- 
face. Hence  it  is  evident  that  through  every  point  of  the  sur- 
face two  rectilinear  elements  can  be  drawn. 

118.  To  represent  a  plane  which  is  tangent  to  a  hyperboloid 
of  revolution  of  one  sheet  at  a  point  of  the  surface. 

Let  the  surface  be  given  as  in  Fig.  65  and  let  P  be  the  given 
point  (Art.  116). 

Analysis.  Draw  the  two  rectilinear  elements  of  the  surface 
which  pass  through  the  given  point.  Since  the  tangents  to 
these  rectilinear  elements  coincide  with  the  elements  them- 
selves, the  elements  determine  the  tangent  plane  to  the  surface 
at  the  given  point.  Either  of  these  rectilinear  elements  with 
the  tangent  to  the  horizontal  circle  of  the  surface  through  the 
given  point  will  also  represent  the  required  tangent  plane  to 
the  surface. 

Construction.  Through  P  draw  the  rectilinear  elements  PC 
and  PO  (Art.  116).  These  elements  determine  the  required 
tangent  plane.  OC  is  the  line  of  intersection  of  this  plane  with 
ihe  plane  of  the  base.  Since  the  horizontal  OC  of  this  plane 
intersects  the  circle  of  the  base  of  the  surface,  it  is  evident  that 
the  tangent  plane  is  also  an  intersecting  plane. 

Since  a  rectilinear  element  of  the  surface,  as  KG,  which 
passes  through  any  point  K  of  the  element  OP  does  not  inter- 
sect the  line  OC,  it  does  not  lie  in  the  tangent  plane  OPC. 
Therefore  OPC  is  not  tangent  to  the  surface  at  the  point  K. 
Hence  a  tangent  plane  to  a  warped  surface  will  contain  a  rec- 
tilinear element  of  the  surface  but  will  be  tangent  to  the  surface 
at  only  one  point  of  that  element. 

HELICOIDS 

119.  A  helicoid  is  the  path  of  a  straight  line  moving  along 
two  helices  having  the  same  pitch  and  lying  on  concentric  cylin- 
ders of  revolution.  The  generatrix  makes  a  constant  angle 
with  the  plane  of  the  bases  of  the  cylinders  and  may  or  may  not 
intersect  their  common  axis. 


102  DESCRIPTIVE   GEOMETRY 

From  the  nature  of  the  motion  it  is  evident  that  no  two  posi- 
tions of  the  generatrix  can  be  brought  into  one  plane  without 
distortion,  therefore  the  surface  is  warped  (Art.  95). 

If  the  generatrix  does  not  cut  the  common  axis  of  the  cylin- 
ders, it  must  remain  a  constant  distance  from  it. 

Each  point  of  the  generatrix  will  generate  a  helix  having  the 
same  axis  and  pitch  as  the  directrices.  In  the  particular  case 
where  the  generatrix  is  tangent  to  one  of  the  directing  helices, 
the  surface  generated  is  a  developable  helicoid  or  convolute 
which  is  a  single  curved  surface. 

When  the  generatrix  is  perpendicular  to  the  axis,  the  surface 
is  a  right  helicoid,  as  in  the  square  screw  thread.  When  the 
generatrix  is  at  an  acute  angle  with  the  axis,  the  surface  is  an 
oblique  helicoid,  as  in  the  triangular  screw  thread. 

120.  To  represent  the  surface.  A  helicoid  is  represented  by 
showing  the  projections  of  its  directrices,  a  few  of  its  rec- 
tilinear elements  and  its  intersection  with  a  horizontal  plane. 

In  Fig.  66,  let  A,  B,  C be  one  of  the  helical  directrices 

and  BBi  the  initial  position  of  the  generatrix.  The  generatrix 
BBi  makes  the  required  angle  oc  with  the  plane  of  the  bases  of 
the  cylinders  and  is  a  constant  distance  d  from  the  axis  MN. 
Any  point  Bj  of  BB^  can  be  selected  as  the  point  which  is  to 
follow  the  other  helical  directrix.  This  directrix  is  then  con- 
structed having  the  same  pitch  and  axis  as  the  other  directrix. 
The  horizontal  projection  of  all  the  elements  of  the  surface  are 
tangent  to  the  circle  which  has  d  for  a  radius  and  mn  for  a  cen- 
ter. The  vertical  projection  of  any  element  CCi  is  found  by 
finding  the  vertical  projections  of  the  points  C  and  C^  where  the 
element  cuts  the  helical  directrices  and  joining  these  points 

with  a  straight  line.     Joining  the  points  A,  Bg,  Cg, in 

which  these  elements  intersect  the  plane  of  the  base  gives  the 
base  of  the  surface. 

The  helicoid  can  be  constructed  by  first  drawing  the  projec- 
tions of  both  helical  directrices.  It  is  then  necessary  to  draw 
the  rectilinear  element  in  such  a  position  that  it  will  touch  both 
helices  and  make  the  required  angle  with  the  plane  of  the. base. 


WARPED  SURFACES 


103 


This  is  accomplished  by  using  any  point  on  either  directrix  as 
the  vertex  of  a  cone  of  revolution  having  its  elements  making 
the  required  angle  with  the  plane  of  the  base.  The  point  in 
which  the  other  directrix  pierces  this  cone  joined  with  the  ver- 


FiG.  66. — Helicoid  and  tangent  plane. 


tex  of  the  cone  gives  one  element  of  the  helicoid.  After  this 
element  is  drawn,  the  other  elements  can  be  drawn  by  the  first 
method  given  above. 

A  point  of  the  surface  is  represented  by  first  representing  an 
element  of  the  surface  and  then  taking  a  point  of  the  element. 

121.  To  represent  a  plane  which  is  tangent  to  a  helicoid  at  a 
point  of  the  surface. 


104  DESCRIPTIVE   GEOMETRY 

Let  the  surface  be  ^iven  as  in  Fig.  66  and  let  Ci  be  the  given 
point,  represented  as  in  Art.  120. 

Analysis,  Construct  a  helix  passing  through  the  point  Ci 
and  lying  on  the  surface  of  the  helicoid.  The  tangent  to  the 
helix  at  the  point  C^  and  the  rectilinear  element  of  the  surface 
through  this  point  vrill  determine  the  tangent  plane. 

Construction.  The  helix  through  the  point  C^  is  constructed 
by  the  method  of  Art.  93.     The  pitch  of  this  helix  must  be  the 

same  a§  the  pitch  of  the  helix  ABC w^hich  represents  the 

helicoid.  The  arc  oc^  is  the  horizontal  projection  of  part  of  this 
helix.  The  straight  line  c^o^  is  equal  to  the  arc  c^^o.  Since  the 
point  Oi  is  on  the  same  level  as  the  point  0,  its  vertical  projec- 
tion o\  is  on  a  line  through  o'  parallel  to  the  ground  line.  Then 
c^Oi  and  c\o\  are  the  projections  of  the  tangent  to  the  helix  at 
the  point  C^  (Art.  94).  cc^  and  dc\  are  the  projections  of  the 
element  of  the  helicoid  passing  through  Ci  (Art.  120).  The 
plane  v^hich  is  determined  by  the  two  lines  CiOi  and  CjC  is  the 
required  tangent  plane.    CgOg  would  be  a  horizontal  of  this  plane. 

The  plane  which  is  tangent  to  the  helicoid  at  any  point  C^  is 
not  tangent  to  the  surface  at  any  other  point  as  C,  Fig.  66. 
The  plane  CgCiOg  is  tangent  to  the  surface  at  Cj.  In  like  man- 
ner, the  plane  of  CCg  and  the  tangent  to  the  helix  ABC at 

C  is  tangent  to  the  surface  at  the  point  C.  These  planes  do  not 
coincide  since  the  tangents  to  the  helices  at  C  and  Ci  are  neither 
parallel  nor  intersecting  lines.  Hence  the  plane  which  is  tan- 
gent to  the  helicoid  at  Ci  is  not  tangent  to  the  surface  at  any 
other  point. 

HYPERBOLIC  PARABOLOIDS 

122.  A  hyperbolic  paraboloid  is  the  path  of  a  straight  line 
moving  along  two  straight  lines  not  in  the  same  plane,  and  re- 
maining parallel  to  a  given  plane. 

The  moving  line  is  called  the  generatrix,  the  two  fixed 
straight  line  the  directrices  and  the  given  plane  the  plane  di- 
rector. Any  position  of  the  generatrix  is  called  an  element  of 
the  surface. 


WARPED   SURFACES  105 

This  surface  has  a  second  rectilinear  generation  in  which  any 
two  rectilinear  elements  of  the  first  generation  may  be  taken  as 
directrices  and  a  plane  which  is  parallel  to  the  first  directrices 
as  a  plane  directer.  It  follows  that  through  any  point  of  a 
hyperbolic  paraboloid,  two  rectilinear  elements  can  always  be 
drawn. 

Any  intersection  of  the  surface  by  a  plane,  which  is  not  a 
straight  line  intersection,  is  a  hyperbola  or  a  parabola,  hence 
the  name  of  the  surface. 

It  is  evident  from  the  nature  of  the  motion  of  the  generating 
line  that  the  surface  is  warped. 

123.  To  represent  the  surface.  The  surface  is  usually  rep- 
resented by  the  projections  of  the  two  rectilinear  directrices, 
the  plane  directer  and  some  of  the  rectilinear  elements. 

To  represent  a  rectilinear  element  of  the  surface.  Let  AB 
and  CD,  Fig.  67,  be  the  rectilinear  directrices  and  let  V  be  the 
plane  directer.  Through  any  point  M  on  the  directrix  AB, 
draw  MN  parallel  to  the  plane  directer  V  and  intersecting  the 
other  directrix  CD  at  N.  mn  and  mV  are  the  horizontal  and 
vertical  projections  respectively  of  a  rectilinear  element  of  the 
surface. 

To  represent  a  point  of  the  surface.  Eepresent  a  rectilinear 
element,  as  MN,  Fig.  67,  then  take  any  point  of  this  element  as 
P.    p  and  p'  are  the  projections  of  the  required  point. 

124.  To  represent  a  plane  which  is  tangent  to  the  surface  at 
a  point  of  the  surface. 

Let  0,  Fig.  67,  be  the  given  point. 

Analysis.  Through  the  given  point  pass  a  plane  parallel  to 
the  elements  of  each  generation.  These  planes  will  intersect 
the  surface  in  the  two  rectilinear  elements  which  determine  the 
tangent  plane  at  the  given  point. 

Construction.  The  plane  through  0  parallel  to  the  elements 
of  the  first  generation  will  cut  the  element  VW  from  the  sur- 
face. Through  0  draw  OG  parallel  to  CD,  and  OU  parallel  to 
AB.  These  lines  will  determine  a  plane  parallel  to  the  elements 
of  the  second  generation.     This  plane  will  intersect  the  element 


106 


DESCRIPTIVE   GEOMETRY 


XY  in  Z.  The  line  joining  Z  with  0  will  be  an  element  of  the 
second  generation.  OW  and  OZ  will  therefore  determine  the 
required  tangent  plane  to  the  surface  at  the  point  0. 


/' 


Fig.  67. — Hyperholic  paraboloid  and  tangent  plane. 


CONOIDS 

125.  A  conoid  is  the  path  of  a  straight  line  moving  along  two 
other  lines,  one  straight  and  the  other  curved,  and  remaining 
parallel  to  a  given  plane. 

To  represent  the  surface.  A  conoid  is  usually  represented  by 
the  projections  of  the  directrices,  the  plane  directer,  and  some 
of  the  rectilinear  elements. 


WARPED  SURFACES 


107 


To  represent  a  rectilinear  element  of  the  surface.    Let  the 
circle  ABCD,  Fig.  68,  and  the  straight  line  MN,  be  the  given 

directrices,  and  let  V  be  the  plane 
directer.  Through  any  point  E  on 
the  curved  directrix  draw  EM  par- 
allel to  the  plane  directer  V,  and 
intersecting  the  rectilinear  direc- 
trix at  M.  em  is  the  horizontal 
and  e'm'  the  vertical  projection  of 
a  rectilinear  element  of  the  sur- 
face. 

To  represent  a  point  of  the  sur- 
face. Represent  a  rectilinear  ele- 
ment, as  EM,  Fig.  68,  and  then 
take  any  point  of  this  element,  as 
0.  0  and  o'  are  the  projections  of 
a  point  of  the  surface. 

126.  To  represent  a  plane  which 
is  tangent  to  the  surface  at  a  given 
point  of  the  surface.  Let  0,  Fig. 
68,  be  the  given  point.  The  tan- 
gent plane  must  contain  the  rectilinear  element  EM  through  the 
given  point.  It  must  also  contain  the  line  OP  tangent  to  the 
elliptical  section  at  the  same  point.  EM  and  OP  will  therefore 
represent  the  required  tangent  plane. 


-Conoid  and  tangent 
plane. 


CYLINDROmS 


127.  A  cylindroid  is  the  path  of  a  straight  line  moving  along 
two  curves,  and  remaining  parallel  to  a  given  plane. 

To  represent  the  surface.  A  cylindroid  is  usually  repre- 
sented by  the  projections  of  the  two  curved  directrices,  the 
plane  directer,  and  some  of  the  rectilinear  elements. 

To  represent  a  rectilinear  element  of  the  surface.  Let  ABC 
and  DEG,  Fig.  69,  be  l^he  curved  directrices,  and  let  H  be  the 


108 


DESCRIPTIVE    GEOMETRY 


plane  directer.  Through  any  point  B  of  one  directrix  draw  a 
straight  line  parallel  to  the  plane  directer  H  and  intersecting 
the  other  directrix  at  E.  h'e'  is  the  vertical  and  te  the  horizon- 
tal projection  of  a  rectilinear  element  of  the  surface. 


Fig.  69. — Cylindroid  and  tangent  plane. 


To  represent  a  point  of  the  surface.  Represent  any  element, 
as  BE,  Fig.  69,  and  then  take  any  point  0  of  this  element,  o  and 
o'  are  the  projections  of  a  point  of  the  surface. 

128.  To  represent  a  plane  which  is  tangent  to  the  surface  at  a 
given  point  of  the  surface.  Let  P,  Fig.  69,  be  the  given  point. 
The  tangent  plane  must  contain  the  rectilinear  element  CP 
through  this  point  and  the  tangent  to  a  plane  section  of  the 
surface  at  P.  NPM  is  a  section  of  the  surface  by  a  plane  per- 
pendicular to  V.  pk  is  the  horizontal  and  p'k'  the  vertical  pro- 
jection of  the  tangent  to  this  curve  at  P.  Then  CP  and  PK 
represent  the  tangent  plane  to  the  surface  at  the  point  P. 


DOUBLE   CURVED  SURFACES  OF  REVOLUTION  109 

DOUBLE  CURVED  SURFACES 

(Only  double  curved  surfaces  of  revolution  will  he  discussed.) 

129.  Some  of  the  simple  double  curved  surfaces  of  revolution 
are  the  following : 

A  sphere  is  the  path  of  a  circle  revolving  about  its  diameter 
as  an  axis. 

A  torus  or  anchor  ring  is  the  path  of  a  circle  revolving  about 
a  straight  line  which  lies  in  the  plane  of  the  circle  but  does  not 
cut  the  circumference. 

An  ellipsoid  of  revolution  is  the  path  of  an  ellipse  revolving 
about  either  axis.  "When  the  ellipse  is  revolved  about  its  major 
axis,  a  prolate  ellipsoid  is  generated;  when  about  its  minor 
axis,  an  oblate  ellipsoid.  These  surfaces  are  sometimes  called 
spheroids. 

A  paraboloid  of  revolution  is  the  path  of  a  parabola  revolving 
about  its  axis. 

A  hyperboloid  of  revolution  of  two  sheets  is  the  path  of  a 
hyperbola  revolving  about  the  axis  which  passes  through  the 
foci. 

These  surfaces  of  revolution  are  usually  represented  with  the 
axis  perpendicular  to  the  horizontal  plane.  The  surface  is  rep- 
resented in  the  horizontal  view  by  its  intersection  with  a  hori- 
zontal plane  or  by  the  projection  of  its  largest  horizontal  circle. 
The  surface  is  represented  in  the  vertical  view  by  the  projection 
of  the  meridian  curve  parallel  to  the  vertical  plane. 

130.  To  represent  a  point  on  a  surface  of  revolution. 

Let  the  surface  be  given  with  the  axis  perpendicular  to  the 
horizontal  plane. 

Assume  the  horizontal  projection  of  the  point  any  place  on 
the  horizontal  projection  of  the  surface.  Through  this  point, 
draw  a  circle  with  center  at  the  horizontal  projection  of  the 
axis.  This  is  the  horizontal  projection  of  a  circle  lying  on  the 
surface.  By  projection,  the  vertical  projection  of  the  circle, 
which  is  a  straight  line  parallel  to  the  ground  line,  is  found. 
The  vertical  projection  of  the  point  is  on  the  vertical  projec- 


110  DESCRIPTIVE   GEOMETRY 

tion  of  the  circle.  These  projections  represent  a  point  on  the 
surface. 

If  the  vertical  projection  of  the  point  is  assumed,  the  vertical 
projection  of  the  circle  must  be  drawn  first,  and  then  the  hori- 
zontal projection  of  the  circle  and  point  found. 

131.  Problems. 

1.  Represent  a  point  on  the  surface  of  a  sphere. 

2.  Represent  a  point  on  the  surface  of  a  prolate  ellipsoid. 

3.  Represent  a  point  on  the  surface  of  an  oblate  ellipsoid. 

4.  Represent  a  point  on  the  surface  of  a  paraboloid  of  revolution. 
6.  Represent  a  point  on  the  surface  of  a  torus. 

132.  Tangent  planes  to  double  curved  surfaces  of  revolution. 

The  tangent  plane  to  a  surface  at  a  point  on  the  surface  is  de- 
termined by  two  straight  lines  tangent  at  this  point  to  two  lines 
of  the  surface.  The  simplest  curves  passing  through  the  point 
on  a  surface  of  revolution  are  usually  the  meridian  line  and  the 
circle  which  is  cut  from  the  surface  by  a  plane  perpendicular 
to  the  axis  of  the  surface.  The  tangent  plane,  therefore,  is 
determined  by  two  straight  lines,  one  tangent  to  the  circular 
section  at  the  given  point  and  the  other  tangent  to  the  meridian 
section  at  that  point. 

133.  To  represent  a  plane  which  is  tangent  to  an  ellipsoid  of 
revolution  at  a  point  of  the  surface. 

Let  the  surface  be  given  as  in  Fig.  70,  and  let  P  be  the  given 
point  represented  by  the  method  of  Art.  130. 

Analysis.  A  tangent  to  the  meridian  curve  at  the  given  point 
and  a  tangent  to  the  circle  of  the  surface  at  this  point  will  rep- 
resent the  tangent  plane  (Art.  132). 

Construction,  prx  is  the  horizontal  and  r'x'  the  vertical  pro- 
jection of  the  circle  of  the  surface  through  the  given  point  P. 
pm  is  the  horizontal  projection  and  p'm'  the  vertical  projection 
of  the  tangent  to  the  circle  at  the  point  P.  ap  is  the  horizontal 
view  of  the  meridian  section  through  P.  If  the  meridian  curve 
through  P  is  revolved  about  the  axis  of  the  surface  until  the 
plane  of  the  curve  is  parallel  to  V,  it  will  have  the  ellipse  rVi/ 
for  its  vertical  projection,  P  moving  to  X.     At  x'  draw  icV  tan- 


DOUBLE  CURVED  SURFACES  OF  REVOLUTION 


111 


gent  to  the  ellipse  xVy',  Then  x'a!  is  the  vertical  projection  of 
the  revolved  position  of  the  tangent  to  the  meridian  section  at 
P.    In  the  counter  revolution  of  the  meridian  plane,  X  moves 


Fig.  70. — Elli'^soid,  of  revolution  and  tan- 
gent 'plane, 

to  P  and  the  point  A,  in  the  axis,  remains  fixed.  Then  ap  is  the 
horizontal  and  a'p'  the  vertical  projection  of  the  tangent  to  the 
meridian  curve  at  P.  PM  and  PA  represent  the  required  plane 
which  is  tangent  to  the  surface  at  the  point  P. 


112  DESCRIPTIVE  GEOMETRY 

134.  Problems. 

1.  Represent  a  plane  which  is  tangent  to  a  sphere,  (a)  at  a  point  of 
the  surface;    (b)  and  which  contains  a  point  outside  the  surface. 

2.  Represent  a  plane  which  is  tangent  to  a  prolate  ellipsoid,  (a)  at  a 
point  of  the  surface;  (b)  and  which  contains  a  point  outside  the  surface. 

3.  Represent  a  plane  which  is  tangent  to  a  paraboloid  of  revolution, 
(a)  at  a  point  of  the  surface;  (b)  and  which  contains  a  point  outside 
the  surface. 

4.  Represent  a  plane  which  is  tangent  to  a  torus,  (a)  at  a  point  of 
the  surface;   (b)  and  which  contains  a  point  outside  the  surface. 

135.  To  represent  a  plane  which  contains  a  given  straight  line 
and  is  tangent  to  a  sphere. 

Let  MN,  Fig.  71,  be  the  given  straight  line  and  let  C  be  the 
center  of  the  given  sphere. 

Analysis.  Assume  that  the  required  plane  is  drawn  through 
the  given  line  and  tangent  to  the  sphere.  Now  if  an  auxiliary- 
plane  be  passed  through  the  center  of  the  sphere  and  perpen- 
dicular to  the  line  MN,  it  will  cut  a  point  from  MN,  a  great 
circle  from  the  sphere  and  a  line  from  the  tangent  plane  which 
will  pass  through  the  point  on  MN  and  be  tangent  to  the  great 
circle  cut  from  the  sphere.  Therefore,  to  make  the  construc- 
tion for  the  tangent  plane,  pass  a  plane  through  the  center  of 
the  sphere  and  perpendicular  to  the  line  MN.  From  the  point 
in  which  this  plane  cuts  MN,  draw  a  tangent  to  the  great  circle 
cut  from  the  sphere.  This  tangent  and  the  line  MN  determine 
the  required  tangent  plane. 

CoTistruction.  HH  and  FF  represent  a  plane  which  contains 
the  center  of  the  sphere  and  is  perpendicular  to  the  liue  MN 
(Art.  50).  D  is  the  point  in  which  the  line  MN  pierces  this 
plane.  When  the  plane  of  HH  and  FF,  which  contains  a  great 
circle  of  the  sphere  and  the  point  D,  is  revolved  about  HH  as  an 
axis  until  it  is  parallel  to  H,  the  point  D  moves  to  Dg  and  the 
great  circle  has  the  same  horizontal  projection  as  the  sphere, 
c^g^z  is  the  horizontal  projection  of  the  revolved  position  of  a 
tangent  from  D  to  the  great  circle.    By  counter  revolution  the 


DOUBLE  CURVED  SURFACES  OF  REVOLUTION 


113 


projections  de  and  d'e'  of  the  required  tangent  are  found. 
DE  and  MN  represent  the  required  tangent  plane. 


Then 


Fig.  71. — Plane  containing  line  and  tangent  to  sphere. 

136.  Problems. 

1.  Represent  a  plane  which  contains  a  line  parallel  to  V  and  is  tan- 
gent to  a  given  sphere. 

2.  Represent  a  plane  which  contains  a  line  parallel  to  H  and  is  tan- 
gent to  a  given  sphere. 

3.  Represent  a  plane  which  contains  a  line  parallel  to  G.  L.  and  is 
tangent  to  a  given  sphere. 

4.  Represent  a  plane  which  contains  a  line  in  a  profile  plane  and  is 
tangent  to  a  given  sphere. 

PROBLEMS 

1.  To  a  given  oblique  cone  with  base  in  a  horizontal  plane,  pass  a 
tangent  plane  which  makes  a  given  angle  with  H. 

2.  To  a  given  oblique  cone  with  base  on  a  plane  parallel  to  V,  pass  a 
tangent  plane  which  makes  a  given  angle  with  V. 


114  DESCRIPTIVE  GEOMETRY 

3.  To  a  given  oblique  cylinder  with  base  in  a  horizontal  plane,  pass  a 
tangent  plane  which  makes  a  given  angle  with  H. 

4.  To  a  given  oblique  cylinder  with  base  on  a  plane  parallel  to  V, 
pass  a  tangent  plane  which  makes  a  given  angle  with  V. 

5.  Find  a  plane  which  contains  the  ground  line  and  is  tangent  to  a 
given  sphere.  Find  the  angles  which  the  plane  makes  with  the  planes 
of  projection. 

6.  Pass  a  plane  tangent  to  a  sphere  and  parallel  to  a  given  plane. 

7.  Inscribe  a  sphere  in  a  given  tetrahedron. 

8.  Having  given  the  axis  of  a  cylinder  of  revolution  and  a  tangent 
plane,  draw  the  projections  of  the  element  of  contact.  (The  tangent 
plane  must  be  parallel  to  the  axis. ) 

9.  Through  a  given  point,  pass  a  plane  tangent  to  a  cylinder  of  rev- 
olution, having  given  the  axis  and  radius  of  the  cylinder,  without  find- 
ing the  projections  of  the  cylinder. 

10.  Having  given  the  radius  and  the  axis  of  a  cylinder  of  revolution, 
to  pass  a  plane  tangent  to  the  cylinder  and  parallel  to  a  given  straight 
line,  without  finding  the  projections  of  the  cylinder. 

11.  A  circle  which  lies  in  a  plane  parallel  with  H  is  the  base  of  a 
cone.  The  vertex  of  the  cone  is  in  G.  L.  Through  some  other  point  on 
Gr.  L.,  pass  a  plane  tangent  to  the  cone.  Show  the  angles  which  the 
plane  makes  with  the  planes  of  projection. 

12.  Find  a  common  normal  to  two  cylinders  of  revolution. 

13.  To  two  given  cylinders,  pass  tangent  planes  which  are  parallel 
to  each  other. 

14.  To  a  given  cone  and  cylinder,  pass  tangent  planes  which  are  par- 
allel to  each  other. 

15.  Two  cones  with  their  bases  on  H  have  the  same  altitude;  pass  a 
plane  tangent  to  each  so  that  the  planes  will  be  parallel. 

16.  Through  a  given  point,  pass  a  plane  tangent  to  two  given  spheres, 

17.  Pass  a  plane  tangent  to  two  given  spheres  and  parallel  to  a  given 
straight  line. 

18.  Pass  a  plane  tangent  to  three  given  spheres. 

19.  Through  a  given  point,  pass  a  plane  which  is  equidistant  from 
three  given  spheres. 

20.  Through  a  given  straight  line,  pass  a  plane  which  is  equidistant 
from  two  given  spheres. 

21.  Pass  a  plane  which  is  equidistant  from  four  given  spheres. 

22.  Find  a  common  tangent  plane  to  a  sphere  and  a  cylinder  of  rev- 
olution. 

23.  Find  a  common  tangent  plane  to  a  sphere  and  a  cone  of  revolution. 


CHAPTER  Y 

PLANE  SECTIONS  AND  DEVELOPMENTS  OF  CURVED 

SURFACES 


137.  The  intersection  of  an  oblique  plane  with  a  curved  sur- 
face which  is  ruled  can  be  found  by  finding  the  points  in  which 
the  rectilinear  elements  of  the  surface  pierce  the  given  oblique 
plane.  These  piercing  points  can  be  found  by  the  usual  method 
of  finding  where  a  straight  line  pierces  a  plane  or  by  using  a 
view  of  the  surface  taken  on  an  auxiliary  plane  of  projection  as 
described  in  Art.  61.  This  auxiliary  plane  is  at  right  angles  to 
a  horizontal  or  a  frontal  of  the  given  oblique  plane.  A  line 
joining  these  piercing  points  in  the  proper  order  is  the  re- 
quired intersection.  The  rectilinear  elements  should  be  taken 
near  enough  together  to  give  the  line  of  intersection  as  accu- 
rately as  desired. 

The  intersection  of  an  oblique  plane  with  a  double  curved 
surface  is  found  by  using  a  system  of  auxiliary  planes.  Each 
of  the  auxiliary  planes  will  cut  a  straight  line  from  the  oblique 
plane  and  a  curved  line  from  the  double  curved  surface  and 
these  lines  will  intersect  in  points  of  the  required  line  of  inter- 
section. The  auxiliary  planes  should  be  so  taken  that  they  will 
cut  simple  curves  from  the  double  curved  surface  and  these 
curves  should  be  in  simple  positions  with  reference  to  the  planes 
of  projection. 

138.  To  find  the  intersection  of  a  right  circular  cylinder  with 
a  given  oblique  plane  and  to  develop  the  surface  of  the  cylinder. 

Let  the  cylinder  be  given  as  in  Fig.  72  and  let  AB  and  AC 
represent  the  oblique  plane. 

Analysis.  Find  the  points  in  which  the  rectilinear  elements 
of  the  cylinder  intersect  the  given  oblique  plane.    A  curved  line 


116 


DESCRIPTIVE   GEOMETRY 


joining  these  points  in  the  proper  order  is  the  required  line  of 
intersection. 

Construction.    Since  the  elements  of  the  cylinder  are  perpen- 
dicular to  H,  the  horizontal  projecting  plane  of  the  element 


Fig.  72. — Plane  section  and  development  of  right  cylinder. 


through  D  may  be  the  horizontal  projecting  plane  of  some  other 
element  as  the  one  through  E.  The  line  of  intersection  of  this 
projecting  plane  with  the  plane  of  AB  and  AC  cuts  the  elements 
D  and  E  at  X  and  Y,  points  on  the  required  line  of  intersection. 
As  many  other  points  as  are  necessary  to  determine  an  accurate 
curve  can  be  found  in  the  same  way. 

139.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  any  point.  Select  the  point  at  which  the  tangent  is  to 
be  drawn  and  then  represent  a  plane  tangent  to  the  cylinder  at 
this  point  (Art.  106).  The  intersection  of  this  plane  with  the 
cutting  plane  is  the  required  tangent  line  (Art.  98).  In  Fig.  72, 
MN  is  the  tangent  to  the  curve  of  intersection  at  the  point  G. 


PLANE  SECTIONS  AND  DEVELOPMENTS  OF  SURFACES  117 

140.  To  develop  the  surface  of  the  cylinder  showing  the  curve 
of  intersection  with  the  oblique  plane.  If  the  cylinder  be  rolled 
on  a  tangent  plane  until  each  element  has  come  into  the  plane, 
the  base  will  develop  into  the  the  straight  line  h^kz.  The  dis- 
tance k202=ko,  Ozd^^od,  etc.  In  the  development,  the  elements 
will  be  perpendicular  to  kzk^  since  they  are  perpendicular  to  the 
plane  of  the  base.  To  get  the  line  of  intersection  on  the  de- 
velopment, lay  off  kzZz^k'z'y  6L^x.^=dJx\  etc.  A  smooth  curve 
passing  through  the  points  z^^  x^,  2/2?  z^  is  the  development  of  the 
line  of  intersection. 

141.  To  find  the  intersection  of  an  oblique  cylinder  with  a 
given  oblique  plane  and  to  develop  the  surface  of  the  cylinder. 

Let  the  cylinder  be  given  as  in  Fig.  73  and  let  AB  and  AC 
represent  the  oblique  plane. 

The  plane  is  taken  perpendicular  to  the  elements  of  the  cylin- 
der. This  is  not  necessary  in  order  to  get  the  intersection  but 
it  is  necessary  for  the  development. 

Analysis.  Find  the  points  in  which  the  rectilinear  elements 
of  the  cylinder  intersect  the  given  oblique  plane.  A  curved  line 
joining  these  points  in  the  proper  order  is  the  required  line  of 
intersection. 

By  following  the  above  analysis,  the  intersection  XYZG  is 
found. 

142.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  any  point.  EG  and  EN  represent  the  tangent  plane  to 
the  cylinder  at  the  point  G  (Art.  106).  The  intersection  of  this 
plane  with  the  plane  of  AB  and  AC  is  the  line  MN.  MN  is  the 
required  tangent  to  the  curve  of  intersection  at  the  point  G. 

143.  To  develop  the  surface  of  the  cylinder.  Since  the  sec- 
tion plane  is  perpendicular  to  the  elements  of  the  cylinder,  it  is 
evident  that  the  line  of  intersection  will  be  a  straight  line  in  the 

development.     The  true  size  of  the  right  section  is  x"\f'z" 

This  is  found  by  revolving  the  section  until  it  is  parallel  to  the 
horizontal  plane  about  AC  as  an  axis.  In  the  development  lay 
off  x^Vz^^x'^',  y2^2=y"^'\  etc.,  along  the  straight  line  w^w^.  On 
a  line  perpendicular  to  W2W2  at  the  point  x^,  lay  off  X20i=x^o; 


118 


DESCRIPTIVE   GEOMETRY 


x^o  being  the  true  length  of  the  element  from  the  plane  of  the 
section  to  the  plane  of  the  base.  In  a  similar  manner,  lay  off 
the  elements  at  2/2?  ^2?  ^tc.  A  smooth  curve  through  the  points 
^2,  2^2j  ^2j is  the  development  of  the  base. 


Fig.  73. — Plane  section  and  development  of  oblique  cylinder. 

144.  To  find  the  intersection  of  an  oblique  cone  with  a  given 
oblique  plane  and  to  develop  the  surface  of  the  cone. 

Let  the  cone  be  given  as  in  Fig.  74  and  let  AB  and  AC  rep- 
resent the  oblique  plane. 

Analysis.  Find  the  points  in  which  the  rectilinear  elements 
of  the  cone  intersect  the  given  oblique  plane.  A  curved  line 
joining  these  points  in  the  proper  order  is  the  required  line  of 
intersection. 

By  following  the  above  analysis,  the  intersection  XYZG  is 
found. 

145.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  any  point.  KG  and  KN  represent  the  tangent  plane  to 
the  cone  at  the  point  G  (Art.  111).    The  intersection  of  this 


PLANE   SECTIONS   AND   DEVELOPMENTS   OF   SURFACES 


119 


plane  with  the  plane  of  AB  and  AC  is  the  line  MN.    MN  is  the 
required  tangent  to  the  curve  of  intersection  at  the  point  G. 

146.  To  develop  the  surface  of  the  cone  showing  the  curve  of 
intersection  with  the  oblique  plane.    Draw  the  elements  V^D, 


Fig.  74. — Plane  section  and  development  of  oblique  cone. 

ViE,  ViO,  etc.  Then  lay  off  the  true  length  v\d\  of  V^D  along 
any  straight  line  as  ^2^2-  With  the  point  V2  as  a  center  and  a 
radius  equal  to  the  true  length  v\e\  of  the  element  V^E,  strike 
an  indefinite  arc.  With  the  point  (^2  ^^  ^  center  and  a  radius 
equal  to  the  rectified  arc  DE,  strike  an  arc  cutting  the  first  arc 
at  ^2.  Then  ^2^2  is  the  element  V^E  laid  off  on  the  development. 
By  using  Vg  as  a  center  and  a  radius  YjO  and  63  as  a  center  and 
a  radius  equal  to  the  rectified  arc  EO,  the  point  O2  is  located. 


120  DESCRIPTIVE  GEOMETRY 

By  continuing  this  process,  the  development  as  shown  in  Fig.  74 
is  found.  Points  on  the  line  of  intersection  are  located  on  the 
development   by  laying  off  V2X2=v\x\,    V2y2=v\y\,   etc.    A 

smooth  curve  through  the  points  Xz,  2/2,  jS2j ^2  is  the  curve 

of  intersection  laid  off  on  the  development.  The  greater  the 
number  of  elements  developed,  the  greater  will  be  the  accuracy 
of  the  development. 

147.  To  find  the  intersection  of  a  warped  surface  with  a  given 
oblique  plane. 

Analysis.  Since  a  warped  surface  is  a  ruled  surface,  it  is 
only  necessary  to  find  the  points  in  which  the  rectilinear  ele- 
ments of  the  surface  pierce  the  given  oblique  plane.  A  line 
joining  these  points  is  the  required  intersection. 

148.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  any  point.  Select  the  point  of  the  curve  at  which  the 
tangent  is  to  be  drawn.  The  intersection  of  the  tangent  plane 
to  the  surface  at  this  point  with  the  plane  of  the  section  is  the 
required  tangent  line. 

A  warped  surface  cannot  be  developed. 

149.  To  find  the  intersection  of  any  surface  of  revolution  with 
a  given  oblique  plane. 

Let  the  surface  be  given  as  in  Fig.  75  and  let  AB  and  AC  rep- 
resent the  oblique  plane. 

Analysis.  Cut  the  surface  and  plane  by  a  system  of  jauxiliary 
planes  perpendicular  to  the  axis  of  the  surface.  The  auxiliary 
planes  will  cut  circles  from  the  surface  which  will  intersect  the 
lines  cut  from  the  given  plane  in  points  of  the  required  curve 
of  intersection. 

Construction.  Since  the  method  is  the  same  for  all  surfaces 
of  revolution,  let  the  construction  be  made  for  the  torus  as  given 
in  Fig.  75.  The  planes,  T,  Tj,  Tg,  etc.,  are  parallel  with  H  and 
cut  lines  from  the  plane  of  AB  and  AC  which  are  parallel  to  a 
horizontal  of  this  plane,  and  intersect  the  circles  cut  from  the 
torus  in  the  points  X,  Xj ;  Y,  Y^ ;  etc.,  points  of  the  required 
curve  of  intersection.     By  continuing  in  this  manner  enough 


PLANE  SECTIONS  OF  SURFACES 


121 


points  can  be  found  to  determine  the  curve  of  intersection  of 
the  given  oblique  plane  with  the  torus. 

150.  To  determine  particular  points  of  the  curve  of  intersec- 
tion. To  find  the  points  where  the  curve  of  intersection  touches 
the  highest  and  lowest  circles  of  the  torus,  use  the  auxiliary 


Fig.  75. — Plane  section  of  surface  of  revolution. 


planes  which  contain  these  circles.  The  vertical  projections 
of  these  circles  are  straight  lines  which  form  part  of  the 
outline  of  the  torus  in  the  vertical  view.  To  find  the  points 
where  the  curve  of  intersection  touches  the  circular  part  of  the 
outline  of  the  surface  in  the  vertical  view,  use  the  auxiliary- 
plane  Tj  which  contains  the  axis  and  is  parallel  to  V.    To  find 


122  DESCRIPTIVE   GEOMETRY 

the  points  where  the  curve  of  intersection  touches  the  circles 
which  represent  the  torus  in  the  horizontal  view,  use  the  auxil- 
iary plane  Tg  which  contains  the  center  of  the  torus  and  is  par- 
allel to  H. 

151.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  a  given  point.  Select  any  point  as  G  at  which  the  tan- 
gent is  to  be  drawn.  The  intersection  MN  of  the  tangent  plane 
to  the  torus  at  this  point  with  the  plane  of  AB  and  AC  is  the 
required  tangent  line. 

152.  To  find  the  line  of  intersection  of  any  surface  with  a 
given  oblique  plane  by  using  an  auxiliary  plane  of  projection. 

Fig.  76  represents  an  oblique  cone  cut  by  the  plane  of  HH  and 
FF.  The  auxiliary  plane  of  projection  is  taken  at  right  angles 
to  the  horizontal  HH  of  the  given  plane.  In  the  auxiliary  view, 
the  given  plane  is  represented  by  the  line  x^z^  and  the  points  in 
which  the  elements  of  the  cone  pierce  this  plane  by  the  points 
^1,  2/i)  ^1'  By  projection  the  horizontal  view  xyz  and  the  ver- 
tical view  x'y'z'  of  the  intersection  are  found. 

In  order  for  the  auxiliary  view  of  the  cutting  plane  to  show 
as  a  straight  line,,  the  auxiliary  plane  of  projection  should  be 
taken  at  right  angles  to  a  horizontal  or  frontal  of  the  given 
plane. 


PLANE  SECTIONS  OP  SURFACES 


123 


-'#.". 


F^G.  76. — 'Plane  section  of  oMique  cone  by  using  an 
auxiliary  plane  of  projection. 


124  DESCRIPTIVE  GEOMETRY 


PROBLEMS 


In  each  of  the  following  problems  draw  a  straight  line  tangent 
to  the  curve  of  intersection  at  some  point  not  on  an  extreme 
element  of  the  surface. 

1.  The  base  of  a  right  cylinder  is  an  ellipse  in  a  horizontal  plane. 
Find  the  intersection  of  this  cylinder  with  a  given  oblique  plane. 

2.  Cut  a  circle  with  diameter  equal  to  the  major  axis  of  the  ellipse 
from  the  cylinder  in  problem  1. 

3.  The  axis  of  a  right  circular  cylinder  makes  60°  with  H  and  is 
oblique  to  V.  Find  the  intersection  of  the  cylinder  with  a  horizontal 
plane. 

4.  The  base  of  an  oblique  cylinder  is  a  circle  in  a  horizontal  plane. 
Cut  a  circle  from  the  cylinder  by  a  plane  which  is  not  parallel  to  the 
base. 

5.  From  a  right  circular  cone  with  base  in  a  horizontal  plane,  cut  an 
ellipse  and  show  the  true  size  of  the  section. 

6.  From  a  right  circular  cone  with  base  in  a  horizontal  plane,  cut  a 
parabola  and  show  the  true  size  of  the  section. 

7.  From  a  right  circular  cone  with  base  in  a  horizontal  plane,  cut  a 
hyperbola  and  show  the  true  size  of  the  section. 

8.  An  oblique  cone  has  a  circle  in  a  horizontal  plane  for  a  base.  Cut 
a  circle  of  given  radius  from  this  cone  by  a  plane  which  is  not  parallel 
to  the  base. 

9.  Is  the  shortest  path  around  a  cone,  following  the  surface,  starting 
at  a  point  and  returning  to  the  same  point,  a  plane  curve? 

10.  Find  the  intersection  of  the  hyperboloid  of  revolution  of  one 
sheet  by  a  plane. 

11.  Find  the  intersection  of  a  helicoid  by  a  plane. 

12.  Find  the  intersection  of  a  hyperbolic  paraboloid  by  a  plane. 

13.  Find  the  intersection  of  a  conoid  by  an  oblique  plane. 

14.  Find  the  intersection  of  a  cylindroid  by  an  oblique  plane. 

15.  Find  the  intersection  of  a  sphere  by  an  oblique  plane. 

16.  Find  the  intersection  of  an  oblate  ellipsoid  of  revolution  by  a 
plane. 

17.  Find  the  intersection  of  a  prolate  ellipsoid  of  revolution  by  a 
given  oblique  plane. 

18.  Find  the  intersection  of  a  torus  by  a  given  oblique  plane. 


CHAPTER  VI 
INTERSECTIONS  OF  CURVED  SURFACES 


153.  To  find  the  line  of  intersection  of  any  two  surfaces.    In 

general,  pass  a  system  of  auxiliary  planes  through  the  surfaces. 
The  planes  will  cut  lines  from  each  surface  and  the  intersections 
of  these  lines  will  be  points  on  the  required  line  of  intersection. 
The  auxiliary  planes  should  be  passed  so  that  they  will  cut  the 
simplest  lines  possible  from  the  given  surfaces. 

Sometimes  the  intersection  can  be  more  easily  found  by  using 
a  system  of  auxiliary  spheres  in  place  of  the  system  of  planes. 
For  example,  if  the  axes  of  two  surfaces  of  revolution  intersect, 
a  system  of  spheres  having  their  centers  at  the  intersection  of 
the  axes  and  radii  of  different  lengths,  will  cut  circles  from  each 
of  the  original  surfaces.  The  intersections  of  these  circles  are 
points  on  the  required  line  of  intersection. 

In  finding  the  line  of  intersection  of  two  curved  surfaces,  the 
points  where  this  line  touches  the  extreme  elements  of  both 
surfaces  should  be  found.  Any  one  of  these  points  can  usually 
be  found  by  taking  an  auxiliary  plane  which  contains  the  ex- 
treme element  through  that  point.  In  some  cases  the  curve  of 
intersection  does  not  touch  the  extreme  elements  of  the  surface. 

154.  To  find  the  line  of  intersection  of  two  oblique  cylinders. 
Let  the  cylinders  be  given  as  in  Fig.  77. 

Analysis.  Pass  a  system  of  auxiliary  planes  parallel  to  the 
elements  of  both  cylinders.  These  planes  will  cut  straight  line 
elements  from  each  cylinder  which  will  intersect  in  points  of 
the  required  curve  of  intersection. 

Construction.  Through  any  point  A  draw  two  lines,  one  par- 
allel to  the  elements  of  one  cylinder  and  the  other  parallel  to 
the  elements  of  the  other  cylinder.     The  plane  of  these  lines 


126 


DESCRIPTIVE  GEOMETRY 


cuts  the  plane  of  the  bases  of  the  cylinders  in  the  line  tt.  Since 
the  auxiliary  planes  are  parallel,  each  of  them  will  cut  a  line 
from  the  plane  of  the  bases  which  is  parallel  to  tt.  The  plane 
Tg,  which  is  tangent  to  one  of  the  cylinders,  cuts  the  element 


Fig.  77. — Intersection  of  two  cylinders. 


BBi  from  that  cylinder  and  the  elements  CCi  and  DD^  from  the, 
other  cylinder.  These  elements  intersect  in  the  points  X  and  T; 
points  of  the  required  line  of  intersection.  Other  points  on  the 
line  of  intersection  are  found  in  the  same  way.  A  curved  line 
joining  in  the  proper  order  the  points  thus  found  is  the  required 
line  of  intersection  of  the  cylinders. 

155.  To  draw  a  tangent  to  the  curve  of  intersection  at  any 
point.     Select  the  point  at  which  the  tangent  is  to  be  drawn  and 


INTERSECTIONS  OF  CURVED  SURFACES  127 

pass  a  plane  tangent  to  each  cylinder  at  this  point.    The  inter- 
section of  these  planes  is  the  required  tangent  to  the  curve. 

156.  To  determine  which  part  of  the  curve  of  intersection  is 
visible.  A  point  on  the  curve  of  intersection  is  visible  in  the 
horizontal  view  if  it  lies  on  an  element  of  each  cylinder  which 
comes  from  the  visible  part  of  the  base  of  the  cylinder  in  that 
view.  In  like  manner,  a  point  is  visible  in  the  vertical  view 
when  it  lies  on  an  element  of  each  cylinder  which  comes  from 
the  visible  part  of  the  base  of  the  cylinder  in  that  view. 

157.  To  find  the  line  of  intersection  of  a  cone  and  cylinder. 
Let  the  surfaces  be  given  as  in  Fig.  78. 

Analysis.  Pass  a  system  of  auxiliary  planes  through  the  ver- 
tex of  the  cone  and  parallel  with  the  elements  of  the  cylinder. 
These  planes  will  cut  from  each  surface  straight  line  elements 
which  will  intersect  in  points  of  the  required  line  of  intersec- 
tion. A  line  through  the  vertex  of  the  cone  and  parallel  with 
the  elements  of  the  cylinder  will  lie  in  all  of  the  auxiliary 
planes. 

Construction.  The  line  Y^X,  through  the  vertex  of  the  cone 
and  parallel  with  the  elements  of  the  cylinder,  lies  in  all  tly' 
auxiliary  planes  and  pierces  the  plane  of  the  bases  at  X,  a  point 
common  to  all  of  the  lines  which  these  planes  cut  from  the  plane 
of  the  bases.  Through  x,  draw  the  lines  xt,  xt^,  etc.,  cutting  the 
bases  of  both  surfaces.  From  the  points  where  these  lines  cross 
the  bases  draw  elements  of  the  surfaces.  These  elements  will 
intersect  in  points  of  the  required  line  of  intersection.  In  this 
manner  the  curve  of  intersection  as  shown  in  Fig.  78'is  found. 

The  visible  part  of  the  curve  is  determined  and  the  tangent 
is  drawn  in  the  same  manner  as  in  Arts.  155  and  156. 

15&  To  determine  in  advance  the  nature  of  the  curves  in 
which  cylinders  and  cones  intersect.  In  the  system  of  auxiliary 
planes  used  in  Fig.  78,  there  are  two  T  and  Tg,  which  are  tan- 
gent to  the  cone,  xt  and  xt^  are  the  intersections  of  these 
planes  with  the  plane  of  the  bases  of  the  surfaces.  The  plane  T 
is  tangent  to  the  cone  on  the  back  side  and  the  plane  Tg  is  tan- 
gent to  the  cone  on  the  front  side.    Both  of  these  tangent  planes 


128 


DESCRIPTIVE   GEOMETRY 


cut  the  cylinder.  It  is  evident  then  that  the  cylinder  extends 
beyond  the  cone  on  both  sides ;  the  cone  running  into  the  cylin- 
der from  one  side  and  out  on  the  other  side,  forming  two  dis- 
tinct curves  of  intersection,  as  is  shown  in  the  figure.     If  one  of 


Fig.  78. — Intersection  of  cone  and  cylinder. 


these  tangent  planes  had  cut  the  base  of  the  cylinder  while  the 
other  did  not,  there  would  have  been  one  continuous  curve  of 
intersection.  If  the  cylinder  had  been  between  these  planes 
and  not  cut  by  either  of  them,  there  would  again  have  been  two 
distinct  curves  of  intersection,  where  the  cylinder  runs  into  the 
cone  from  one  side  and  where  it  runs  out  on  the  other.     In  the 


INTERSECTIONS  OF  CURVED  SURFACES  129 

case  where  the  planes  are  tangent  to  both  cone  and  cylinder, 
the  two  curves  of  intersection  just  touch  each  other. 

The  same  method  may  be  used  to  determine  the  nature  of  the 
curve  of  intersection  of  two  cylinders  or  of  two  cones. 

159.  To  find  the  line  of  intersection  of  two  cones. 

Analysis.  Pass  a  system  of  auxiliary  planes  through  the  ver- 
tices of  both  cones.  These  planes  will  cut  straight  lines  from 
each  cone  which  will  intersect  in  points  of  the  required  line  of 
intersection.  A  line  joining  the  vertices  of  the  cones  will  lie  in 
all  of  the  auxiliary  planes  and  will  pierce  the  plane  of  the  bases 
in  a  point  common  to  all  the  lines  which  these  planes  cut  from 
the  plane  of  the  bases. 

Let  the  construction  be  made  for  finding  the  curve  of  inter- 
section. Also  determine  which  part  of  the  curve  is  visible  and 
draw  a  tangent  to  the  curve  by  methods  previously  given. 

160.  To  find  the  line  of  intersection  of  a  sphere  and  a  cone  or 
a  sphere  and  a  cylinder. 

Analysis  1,  If  the  cone  or  cylinder  has  a  circle  for  a  base, 
the  auxiliary  planes  can  be  passed  parallel  to  the  base.  Each 
plane  will  cut  a  circle  from  the  sphere  and  a  circle  from  the 
cone  or  cylinder.  The  intersections  of  these  circles  are  points 
on  the  required  line  of  intersection. 

Analysis  2.  Take  the  horizontal  or  vertical  projecting  planes 
of  the  rectilinear  elements  of  the  cone  or  cylinder.  These  planes 
will  cut  circles  from  the  sphere  which  will  intersect  the  ele- 
ments cut  from  the  cone  or  cylinder  in  points  of  the  required 
line  of  intersection.  To  find  these  points,  it  may  be  necessary 
to  revolve  the  projecting  planes  until  they  are  parallel  to  one 
of  the  planes  of  projection. 

Let  the  construction  be  made  in  accordance  with  the  above 
analysis. 

161.  To  draw  a  straight  line  tangent  to  the  curve  of  intersec- 
tion at  any  point.  The  required  tangent  will  be  the  line  of  in- 
tersection of  two  planes,  one  tangent  to  the  cone  or  cylinder 
and  the  other  tangent  to  the  sphere  at  the  given  point. 


130  DESCRIPTIVE  GEOMETRY 


PROBLEMS 

1.  Find  the  intersection  of  an  oblique  cone  and  a  right  circular  cylin- 
der.   The  bases  of  both  surfaces  are  circles  on  a  horizontal  plane. 

2.  Find  the  intersection  of  a  right  circular  cone  with  a  sphere.  The 
base  of  the  cone  is  on  a  horizontal  plane  and  the  center  of  the  sphere 
not  on  the  axis  of  the  cone. 

3.  Find  the  intersection  of  a  right  circular  cylinder  with  a  right 
circular  cone.  The  axes  of  both  surfaces  are  perpendicular  to  H  but 
do  not  coincide. 

4.  Find  the  intersection  of  a  sphere  with  a  right  circular  cylinder. 
The  axis  of  the  cylinder  is  perpendicular  to  H  but  does  not  pass 
through  the  center  of  the  sphere. 

5.  Find  the  intersection  of  a  right  circular  cone  with  a  right  circular 
cylinder.  The  base  of  the  cone  is  on  a  horizontal  plane  and  the  axis 
of  the  cylinder  is  parallel  to  the  ground  line.  Draw  a  tangent  to  the 
curve  of  intersection. 

6.  Find  the  intersection  of  a  triangular  prism  with  a  sphere. 

7.  Find  the  line  of  intersection  of  a  cone  with  a  triangular  prism. 

8.  Find  the  line  of  intersection  of  the  two  given  oblique  cylinders, 
which  have  their  bases  in  a  horizontal  plane,  and  draw  a  tangent  to  the 
curve  at  some  point  of  the  curve  not  on  an  extreme  element  of  either 
surface. 


Cylinder  A  < 


Center  of  lower  base,  A(3i,  X,  — 4) 
Center  of  upper  base,  0(— 3^,  —If,  0) 
Major  axis  is  3^"  long  and  is  parallel  to  V 
Minor  axis  is  2^"  long 


r  Center  of  lower  base,  B(— 3^,  Y,  — 4) 
Cy'linder  B  J   Center  of  upper  base,  C(i,  — 4i,  0) 
1   Radius  of  base  If" 

Note. — Take  the  origin  at  the  center  of  a  ll''xl5"  sheet,  with  the 
ground  line  parallel  to  the  longer  side  and  passing  through  the  center. 
X  may  vary  from  — 1^"  to  —ZV  and  Y  may  vary  from  —2"  to  —3". 

9.  Find  the  line  of  intersection  of  the  given  oblique  cylinder  and  the 
given  oblique  cone,  which  have  their  bases  in  a  horizontal  plane,  and 
draw  a  tangent  to  the  curve  at  some  point  of  the  curve  not  on  an  ex- 
treme element  of  either  surface. 


PROBLEMS  131 

{Center  of  the  lower  base,  A(i,  X,  — 3^) 
Center  of  the  upper  base,  C(— 3,  — li,  0) 
Radius  of  the  base,  li" 

r  Center  of  the  base,  B(— 3i,  Y.  — 3i) 
Cone  B      J   Vertex,  Y^dh  ~5i,  li) 
I  Radius  of  the  base,  If" 

Note.— Take  the  origin  at  the  center  of  a  ll"xl5''  sheet,  with  the 
ground  line  parallel  to  the  longer  side  and  passing  through  the  center. 
X  may  vary  from  —If'  to  — 3i"  and  Y  may  vary  from  —2"  to  — 3J". 

10.  Find  the  line  of  intersection  of  two  oblique  cones,  which  have 
their  bases  in  a  horizontal  plane,  and  draw  a  line  tangent  to  the  curve 
of  intersection  at  some  point  of  the  curve  not  on  an  extreme  element 
of  either  surface. 

r  Center  of  the  base,  A(4i,  X,  — 4f) 
Cone  A         J   Vertex  of  the  cone,  V^(l,  — 2|,  —2) 
Radius  of  the  base,  1^" 


r  Center  of  the  base,  B(0,  Y,  — 4|) 
Cone  B         <   Vertex  of  the  cone,  V^CB^,  — 5i,  0) 
(^Radius  of  the  base.  If" 

Note. — Take  the  origin  at  the  center  of  a  ll"xl5"  sheet  with  the 
ground  line  parallel  to  the  longer  side  and  passing  through  the  center. 
X  may  vary  from  — If"  to  — 3i"  and  Y  may  vary  from  — 2"  to  — 3". 

11.  Find  the  line  of  intersection  of  the  two  given  cones,  which  have 
their  bases  in  a  horizontal  plane,  and  draw  a  line  tangent  to  the  curve 
of  intersection  at  some  point  of  the  curve  not  on  an  extreme  element  of 
either  surface. 

r  Center  of  the  base,  A(0,  — 3i,  — 6f ) 
Cone  A         J   Vertex  of  the  cone,  VJO,  —31,  0) 
[^  Radius  of  the  base,  3" 

/Center  of  the  base,  B(i,  —4,  — 6f) 
Vertex  of  the  cone,  V^C— 3i,  —3,  — li) 
Radius  of  the  base,  2J" 


Take  origin  at  the  center  of  a  ll"xl5"  sheet,  with  the  ground  line 
passing  through  the  center  and  parallel  to  the  shorter  side  of  the  sheet. 

12.  Find  the  line  of  intersection  of  a  torus  and  an  oblique  cylinder. 


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